Euclid - a short biography. Ancient Greek mathematician Euclid: biography of a scientist, discoveries and interesting facts

Name: Euclid (Euclid)

Years of life: about 325 BC NS. - 265 BC NS.

State: Ancient Greece

Field of activity: Science, Mathematics, Geometry

Everyone knows that science was not invented yesterday - even in ancient times, outstanding minds discovered various theorems, theories, created new elements. Mathematics and astronomy were especially honored. The Egyptians also succeeded in these sciences.

Now it is impossible to imagine mathematics without a theorem, without the famous discovery of Archimedes in the bathroom. There was another Greek who made a tangible contribution to science in general. His name is Euclid.

Euclid (325 BC - 265 BC) - Greek mathematician. He is considered the "father of geometry". His textbook, The Elements, remained a highly sought after and accurate textbook in mathematics until the late 19th century and is one of the most widely published books in the world. But what about the author himself? Unfortunately, not much. Information about his life is extremely scarce and often implausible.

Biography of Euclid

Euclid was born in the middle of the 4th century BC and lived in Alexandria, in the territory; the peak of his creative activity came during his reign (323-283 BC), and his name Euclid means "famous, glorious." In some sources, he is also referred to as Euclid of Alexandria.

It is likely that Euclid worked with a team of mathematicians in Alexandria, and he obtained his degree through his mathematical work. Some historians believe that Euclid's work may have been the result of multiple authors, but most agree that one person - Euclid - was the main author.

It is likely that Euclid attended the Academy in Athens, and most of his knowledge came from there. It was there that he first became acquainted with mathematics, namely with one part of it - geometry.

Contemporaries described him as a kind, pleasant person in communication. For example, the historian Papp writes that Euclid was

“... the most fair and benevolent towards everyone who has been able to advance mathematics in any way. He responded carefully so as not to hurt him in any way. And although he was a great scientist, he never boasted himself. "

The mathematician's personal life is unknown - he devoted almost all his time to science.

Euclid's postulates

His main book The Elements (originally written in ancient Greek) became the basic work of important mathematical teachings. It is divided into 13 separate books.

Books from the first to the sixth are devoted to the geometry of the plane.
Books seven-nine deal with number theory
Book eight about geometric progression
Book Ten is about Irrational Numbers
Books eleven to thirteen represent three-dimensional geometry (stereometry).

Euclid's genius was to take many different elements of mathematical ideas into circulation and combine them into one logical, coherent format.

Euclid's lemma, which states that a fundamental property of primes is that if a prime divides the product of two numbers, it must divide at least one of those numbers.

Euclid's Algorithm

Using Euclid's lemma, this theorem states that every integer greater than one is either a prime in itself or a product of primes, and that there is a certain order of primes.

"If two numbers, multiplying one by the other, make up a number, and any number that is divisible by their product will also be divisible by each of the original numbers."

The Euclidean algorithm is an efficient method for calculating the greatest common divisor (GCD) of two numbers, the largest number that divides both of them, leaving no remainder.

Euclidean geometry

Euclid described a system of geometry related to shape, relative position and properties of space. His work is known as Euclidean geometry. Space is assumed to have a dimension equal to three.

Sometimes his work "Elements" is compared with the Bible - in the sense that his work has been translated into many languages and literally became the reference book of many scientists and mathematicians of subsequent centuries.

Besides geometry, Euclid explored other branches of mathematics. However, it is worth recognizing that Euclid's contribution to science is enormous - without him, probably, mathematics would not have been able to reveal itself so much to scientists. His name is inextricably linked with geometry, the study of space.

Euclid (Eukleides)

3rd century BC NS.

Archimedes' main work is "Beginnings" (lat. Elementa) - contains a presentation of planimetry, stereometry and a number of questions of number theory (for example, Euclid's algorithm); consists of 13 books, to which are added two books about five regular polyhedra, sometimes attributed to Hypsicles of Alexandria. In the Elements, he summed up the previous development of Greek mathematics and laid the foundation for the further development of mathematics. For more than two millennia, the Euclidean Principles remained the main work in elementary mathematics.

Among other mathematical works of Euclid, it should be noted "On the division of figures", preserved in the Arabic translation, four books "Conical sections", the material of which was included in the work of the same name by Apollonius of Perga, as well as "Porisms", an idea of which can be obtained from the "Mathematical collection" Pappa of Alexandria.

In the writings of Euclid, a systematic exposition of the so-called. Euclidean geometry, the system of axioms of which is based on the following basic concepts: point, line, plane, motion and the following relations: "a point lies on a straight line on a plane", "a point lies between two others". In the modern presentation, the system of axioms of Euclidean geometry is divided into the following five groups.

I. Combination axioms. 1) Through every two points, you can draw a straight line and, moreover, only one. 2) Each line contains at least two points. There are at least three points that are not collinear. 3) Through every three points that do not lie on one straight line, you can draw a plane and, moreover, only one. 4) Each plane has at least three points and there are at least four points that do not lie in the same plane. 5) If two points of a given line lie on a given plane, then the line itself lies on this plane. 6) If two planes have a common point, then they have one more common point (and, therefore, a common straight line).

II. Order axioms. 1) If point B lies between A and C, then all three lie on one straight line. 2) For each point A, B, there is a point C such that B lies between A and C. 3) Of the three points of a straight line, only one lies between two others. 4) If a straight line intersects one side of a triangle, then it intersects another side of it or passes through a vertex (segment AB is defined as the set of points lying between A and B; the sides of the triangle are determined accordingly).

III. Axioms of motion. 1) The movement puts in correspondence to points points, straight lines, planes of a plane, keeping the belonging of points to straight lines and planes. 2) Two successive movements give again movement, and for every movement there is an opposite. 3) If points are given A, A " and half-plane a, a"bounded by extended half-lines a, a " that come from points A, A ", then there is a movement, and, moreover, the only one that translates A, a, a v A ", a", a "(half-line and half-plane are easily defined based on the concepts of combination and order).

IV. Continuity axioms. 1) Archimedes' axiom: any segment can be overlapped by any segment, postponing it on the first a sufficient number of times (the postponing of the segment is carried out by movement). 2) Cantor's axiom: if you are given a sequence of segments embedded in one another, then they all have at least one common point.

V. Axiom of Euclid's parallelism. Through point A out of line a in the plane passing through A and a, you can draw only one straight line that does not intersect a.

The emergence of Euclidean geometry is closely related to visual representations of the world around us (straight lines - stretched threads, rays of light, etc.). The long process of deepening our understanding has led to a more abstract understanding of geometry. The discovery by N.I. Lobachevsky of geometry, different from Euclidean, showed that our ideas about space are not a priori. In other words, Euclidean geometry cannot claim to be the only geometry that describes the properties of the space around us. The development of natural science (mainly physics and astronomy) has shown that Euclidean geometry describes the structure of the space around us only with a certain degree of accuracy and is not suitable for describing the properties of space associated with the movement of bodies with speeds close to light. Thus, Euclidean geometry can be considered as the first approximation for describing the structure of real physical space.

Almost nothing is known about Euclid's life. The first commentator of the "Principles" Proclus (V century AD) could not indicate where and when Euclid was born and died ...

Some biographical data have been preserved on the pages of an Arab manuscript of the 12th century: "Euclid, son of Naukrat, known as" Geometer ", a scientist of the old times, Greek by origin, Syrian by domicile, originally from Tire."

Tsar Ptolemy I attracted scientists and poets to Egypt, creating for them a temple of muses - Museion. Among the invited scientists was Euclid, who founded in Alexandria - the capital of Egypt - a mathematical school and wrote for her students his fundamental work, united under the general title "Beginnings". It was written around 325 BC.

The Beginnings consists of thirteen books, organized according to a single logical scheme. Each of the thirteen books begins with the definition of concepts (point, line, plane, figure, etc.) that are used in it, and then, based on a small number of basic provisions (5 axioms and 5 postulates), taken without proof, the whole system is built geometry.

Books I-IV covered geometry, their content went back to the works of the Pythagorean school. Book V developed the doctrine of proportions. Books VII-IX contained the doctrine of numbers, representing the development of the Pythagorean primary sources. Books X-XII contain definitions of areas in plane and space (stereometry), the theory of irrationality (especially in X book); Book XIII contains studies of regular bodies.

The "principles" of Euclid are a presentation of the geometry that is known to this day under the name of Euclidean geometry. It describes the metric properties of space, which modern science calls Euclidean space. This space is empty, boundless, isotropic, having three dimensions. Euclid gave mathematical definiteness to the atomistic idea of empty space in which atoms move. Euclid's simplest geometric object is a point, which he defines as something that has no parts. In other words, a point is an indivisible atom of space.

The doctrine of parallel straight lines and the famous fifth postulate ("If a straight line falling on two straight lines forms internal angles that are less than two straight lines on one side, then these two straight lines extended indefinitely will meet on the side where the angles are less than two straight lines") determine the properties of the Euclidean space and its geometry, different from non-Euclidean geometries.

Over the course of four centuries, "Beginnings" were published 2500 times: on average, 6-7 editions were published annually. Until the 20th century, the book was considered the main textbook on geometry not only for schools, but also for universities.

Partly preserved, partly reconstructed later mathematical works belong to Euclid. It was he who introduced the algorithm for obtaining the greatest common divisor of two arbitrarily taken natural numbers and the algorithm called "the Eratosthenes count" - for finding the prime numbers from a given number.

Euclid laid the foundations of geometric optics, which he outlined in the works "Optics" and "Catoptrika". In Euclid, we also find a description of a monochord - a one-string device for determining the pitch of a string and its parts. The invention of the monochord was important for the development of music. Gradually, instead of one string, two or three were used. This was the beginning of the creation of keyboard instruments, first the harpsichord, then the piano.

Of course, all the features of the Euclidean space were not immediately discovered, but as a result of centuries of work of scientific thought, but the starting point of this work was the "Principles" of Euclid. Knowledge of the foundations of Euclidean geometry is now a necessary element of general education throughout the world.

Euclid (aka Euclid) is an ancient Greek mathematician, the author of the first theoretical treatise on mathematics that has come down to us. Biographical information about Euclid is extremely scarce. It is only known that the teachers of Euclid in Athens were the students of Plato, and during the reign of Ptolemy I (306-283 BC) he taught at the Alexandrian Academy. Euclid is the first mathematician of the Alexandrian school. Euclid is the author of a number of works on astronomy, optics, music, and others. Arab authors ascribe to Euclid various treatises on mechanics, including works on weights and on the determination of specific gravity. Died Euclid between 275 and 270 BC. NS.

The beginnings of Euclid

Euclid's main work is called Beginnings. Books with the same title, in which all the basic facts of geometry and theoretical arithmetic were consistently set forth, were previously compiled by Hippocrates of Chios, Leon and Theudy. However, the Principles of Euclid supplanted all these works from everyday life and for more than two millennia remained the basic textbook of geometry. In creating his textbook, Euclid incorporated into it much of what was created by his predecessors, processing this material and bringing it together.

The 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 set basic constructions (for example, “it is required that a straight line can be drawn through any two points”), and axioms - general inference rules when operating with quantities (eg, “if two quantities are equal to the third, they are equal between themselves").

Book I studies the properties of triangles and parallelograms; this book is crowned with the famous Pythagorean theorem for right-angled triangles. Book II, dating 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 use the works 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. VII-IX books are devoted to the theory of numbers and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. In these books, theorems on proportions and geometric progressions are considered, a method is introduced for finding the greatest common divisor of two numbers (now known as Euclid's algorithm), even perfect numbers are constructed, and the infinity of the set of primes is proved. In Book X, which is the most voluminous and complex part of the Principles, a classification of irrationalities is constructed; it is possible that its author is Theetetus of Athens. XI book contains the basics of stereometry. In the XII book, using the method of exhaustion, theorems are proved about the ratios of the areas of circles, as well as the volumes of pyramids and cones; the author of this book is admittedly Eudoxus of Cnidus. Finally, Book XIII is devoted to the construction of five regular polyhedra; it is believed that some of the buildings were designed by Theetetus of Athens.

In the manuscripts that have come down to us, two more are added to these thirteen books. The XIV book belongs to the Alexandrian Hypsicles (c. 200 BC), and the XV book was created during the life of Isidore of Miletus, the builder of the church of St. Sophia in Constantinople (early 6th century AD).

The beginnings provide a common basis for subsequent geometric treatises by Archimedes, Apollonius, and other ancient authors; the proposals proven in them are considered to be generally known. Comments on the Principles in antiquity were composed by Heron, Porfiry, Papp, Proclus, Simplicius. Proclus's commentary on Book I, as well as Pappus's commentary on Book X (in Arabic translation) have survived. From ancient authors, the commentary tradition passes to the Arabs, and then to Medieval Europe.

In the creation and development of modern science, the Beginnings also played an important ideological role. They remained a model of a mathematical treatise, rigorously and systematically setting out the main provisions of this or that mathematical science.

Euclid's second work after the "Beginnings" is usually called "Data" - an introduction to geometric analysis. Euclid also owns "Phenomena" devoted to elementary spherical astronomy, "Optics" and "Catoptrica", a small treatise "Sections of the Canon" (contains ten problems on musical intervals), a collection of problems on dividing the areas of figures "On divisions" (reached us in Arabic translation). The exposition in all these works, as in the "Elements", is subject to strict logic, and the theorems are deduced from precisely formulated physical hypotheses and mathematical postulates. Many of Euclid's works have been lost; we know about their existence in the past only through references in the works of other authors.

Euclid, son of Naukrat, known under the name of "Geometer", a scientist of the old times, Greek by origin, Syrian by domicile, originally from Tire. "

One of the legends says that King Ptolemy decided to study geometry. But it turned out that this is not so easy to do. Then he called Euclid and asked him to show him the easy way to mathematics. “There is no royal road to geometry,” the scientist answered him. So, in the form of a legend, this became a popular expression.

Tsar Ptolemy I, in order to exalt his state, attracted scientists and poets to the country, creating for them a temple of muses - Museion. There were study rooms, botanical and zoological gardens, an astronomical office, an astronomical tower, rooms for secluded work and, most importantly, a magnificent library. Among the invited scientists was Euclid, who founded a mathematical school in Alexandria, the capital of Egypt, and wrote his fundamental work for her students.

It was in Alexandria that Euclid founds a mathematical school and wrote a large work on geometry, united under the general title "Beginnings" - the main work of his life. It is believed to have been written around 325 BC.

The predecessors of Euclid - Thales, Pythagoras, Aristotle and others did a lot for the development of geometry. But these were all separate fragments, not a single logical scheme.

Usually it is said about Euclid's "Beginnings" that after the Bible it is the most popular written monument of antiquity. The book has a very remarkable history. For two thousand years, it was a handbook for schoolchildren, used as an initial geometry course. The Beginnings were extremely popular, and many copies were made of them by hardworking scribes in different cities and countries. Later, the "Beginnings" from papyrus were transferred to parchment, and then to paper. Over the course of four centuries, Beginnings were published 2,500 times: on average, 6-7 editions were published annually. Until the 20th century, the book "Beginnings" was considered the main textbook on geometry not only for schools, but also for universities.

The "beginnings" of Euclid were thoroughly studied by the Arabs and later by European scientists. They have been translated into major world languages. The first originals were printed in 1533 in Basel It is curious that the first translation into English, dating back to 1570, was made by Henry Billingway, a London merchant

Knowledge of the foundations of Euclidean geometry is now a necessary element of general education throughout the world.

In arithmetic, Euclid made three significant discoveries. First, he formulated (without proof) a division theorem with remainder. Second, he came up with the "Euclidean algorithm" - a quick way to find the greatest common divisor of numbers or the common measure of segments (if they are commensurable). Finally, Euclid was the first to study the properties of primes - and proved that their set is infinite.

Euclid or Euclid(Old Greek. Εὐκλείδης , from "good glory", heyday - about 300 BC. BC) - an ancient Greek mathematician, the 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 NS.

Biography

It is customary to attribute to the most reliable information about the life of Euclid the little that is given in Proclus's comments to the first book Started Euclid (although it should be borne in mind that Proclus lived almost 800 years after Euclid). Noting that "mathematicians who wrote on the history of mathematics" did not bring the presentation of the development of this science to the time of Euclid, Proclus points out that Euclid was younger than the Platonic circle, but older than Archimedes and Eratosthenes, "he lived at 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 way to geometry. "

Additional touches to Euclid's portrait can be found in Papp and Stobey. Papp reports that Euclid was gentle and kind to everyone who could contribute at least in the slightest degree to the development of the mathematical sciences, and Stobey narrates another anecdote about Euclid. Starting to study geometry and analyzing the first theorem, one young man asked Euclid: "And what benefit will I get from this science?" Euclid called a slave and said: "Give him three obols, since he wants to profit from his studies." The historicity of the story is questionable, since a similar story is told about Plato.

Some modern authors interpret Proclus's statement - Euclid lived at 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 concept was established in Europe in the 17th century, while medieval authors identified Euclid with the philosopher Euclid of Megar, a student of Socrates.

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

Euclid, son of Naukrat, known as "Geometer", a scientist of the old times, Greek by origin, Syrian by domicile, originally from Tire ...

The formation of Alexandrian mathematics (geometric algebra) as a science is also associated with the name of Euclid. In general, the amount of data about Euclid is so scanty that there is a version (though not widely spread) that we are talking about the collective pseudonym of a group of Alexandrian scientists.

« Beginnings»Euclid

The main work of Euclid is called Beginnings. Books with the same title, in which all the basic facts of geometry and theoretical arithmetic were consistently set forth, were previously compiled by Hippocrates of Chios, Leon and Theudy. but Beginnings Euclid supplanted all these writings from everyday life and for more than two millennia remained the basic textbook of geometry. In creating his textbook, Euclid incorporated into 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 set basic constructions (for example, “it is required that a straight line can be drawn through any two points”), and axioms - general inference rules when operating with quantities (eg, “if two quantities are equal to the third, they are equal between yourself ").

Euclid opens the gates of the Garden of Mathematics. Illustration from the treatise of Niccolo Tartaglia "New Science"

Book I studies the properties of triangles and parallelograms; this book is crowned with the famous Pythagorean theorem for right-angled triangles. Book II, dating 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 use the works 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. VII-IX books are devoted to the theory of numbers and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. In these books, theorems on proportions and geometric progressions are considered, a method is introduced to find the greatest common divisor of two numbers (now known as Euclid's algorithm), even perfect numbers are constructed, and the infinity of the set of primes is proved. In the X book, which is the most voluminous and complex part Started, a classification of irrationalities is being built; it is possible that its author is Theetetus of Athens. XI book contains the basics of stereometry. In the XII book, using the method of exhaustion, theorems are proved about the ratios of the areas of circles, as well as the volumes of pyramids and cones; the author of this book is admittedly Eudoxus of Cnidus. Finally, Book XIII is devoted to the construction of five regular polyhedra; it is believed that some of the buildings were designed by Theetetus of Athens.

Beginnings provide a common basis for subsequent geometric treatises by Archimedes, Apollonius and other ancient authors; the proposals proven in them are considered to be generally known. Comments on Beginnings in antiquity they were Heron, Porfiry, Pappus, Proclus, Simplicius. Proclus's commentary on Book I, as well as Pappus's commentary on Book X (in Arabic translation) have survived. 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, rigorously and systematically setting out the main provisions of this or that mathematical science.

Other works of Euclid

Other works of Euclid have survived:

Data (δεδομένα ) - about what is needed to set the shape;
About division (περὶ διαιρέσεων ) - partially preserved and only in the Arabic translation; gives the division of geometric shapes into parts equal or consisting of each other in a given ratio;
Phenomena (φαινόμενα ) - applications of spherical geometry to astronomy;
Optics (ὀπτικά ) - about rectilinear light propagation.

Known by short descriptions:

Porisms (πορίσματα ) - about the conditions that determine the curves;
Conical sections (κωνικά );
Surface 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

Texts and translations

Old Russian translations

Euclidean elements from twelve nephton books were selected and abbreviated into eight books through professor of mafematics A. Farhvarson. / Per. from lat. I. Satarova. SPb., 1739.284 p.
Elements of geometry, that is, the first foundations of the science of measuring length, consisting of axes Euclidean books. / Per. from French N. Kurganova. SPb., 1769.288 p.
Euclidean elements of eight books, namely: 1st, 2nd, 3rd, 4th, 5th, 6th, 11th and 12th. / Per. from Greek. SPb.,