(330 BC-260 BC)

ancient greek mathematician

Euclid was born in 330 BC. in the small town of Tire, near Athens. History has not left a detailed description of the life of one of the most famous mathematicians of all times and peoples.

Once King Ptolemy asked Euclid if there was another, not such a difficult way of learning geometry than the one that the scientist outlined in his "Elements". Euclid replied: "O king, there are no royal roads in geometry."

For a long time, scientists believed that there was no specific historical person, that under the name of Euclid there was a group of mathematicians, something like our contemporary Bourbaki, by the way, a great teacher. However, in the 12th century manuscript in Arabic we read: "Euclid, son of Naukrat, son of Zenarch, known as Geometer, a scientist of the old times, Greek by origin, Syrian by origin, originally from Tire."

Euclid, a student of Plato, at the invitation of King Ptolemy moved to Alexandria, where the famous scientific center with the Alexandrian Library was located.

The famous work "Beginnings" (Stoicheia) made his name immortal. The Beginnings consists of thirteen books. Other works of Euclid are less known and less in volume. These are, first of all, “Data”, “Optics”, “On the division of figures”, “False conclusions” (lost), “Section of the Canon”, “Phenomena”.

This is a great encyclopedic teacher who taught in Alexandria, in Museion. This is a real palace of science with a library, an astronomical observatory, a botanical garden, a zoo. Famous scientists were invited to Museion, they conducted scientific work here, and received a good reward. The work of a scientist has become a profession. Euclid teaches geometry, arithmetic and astronomy at the Museion.

Euclid's "beginnings" constitute an entire epoch in elementary geometry. This is a great work. The scientist presents geometry as a chain of rigorous logical inferences, theorem proving based on definitions, postulates and axioms. The original of the Beginnings has not reached us, since the manuscript was kept in the Alexandrian library, which later perished. In the Elements, Euclid outlined the results obtained by his predecessors, the great mathematicians. This required a pedagogical talent and the genius of a systematizer.

What scientific goals did the scientist set for himself, generalizing the experience of famous mathematicians? These goals are three: to outline the theory of relations of the great Eudoxus (406-355 BC), the theory of irrational Tietetus (IV century BC), Plato's theory of five regular bodies (429-348 BC) .). The first four books of the "Elements" are devoted to planimetry, the fifth and sixth - the theory of relations of Eudoxus. Then comes geometry in space, solid angles, volumes of bodies, the theory of numbers is presented.

In the "Beginnings", Eudoxus' algorithm for finding the greatest common divisor is given. Here the ideas of the Architus from Tarenta (428-365 BC) are presented. Finally, after stereometry, Euclid expounds the theory of Eudoxus' exhaustion and applications to the area of ​​a circle and volume of a sphere, cone and pyramid. Euclid expounds the theory of five Platonic solids according to Tietetus.

The famous V axiom of Euclid (V postulate) occupies a special place in the "Elements". Numerous attempts in the 19th century to "correct" a scientist, to make a theorem out of this axiom ended in failure.

His "Beginnings" is an example of a deductive presentation of geometry, algebraic conclusions are made in a geometric style. Subsequently, geometry developed, non-Euclidean geometry appeared, geometry became an experimental science in physics. But the prerequisites for this development were precisely the works of the great Euclid.

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 “those 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 path 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. Having started to study geometry and having analyzed the first theorem, one young man asked Euclid: "And what benefit will I get from this science?" Euclid called the 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 of Socrates, the philosopher Euclid of Megar.

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 Feudy. 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 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 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.

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).

Beginnings provide a common basis for subsequent geometric treatises by Archimedes, Apollonius and other ancient authors; the proposals proved in them are considered to be generally known. Comments to 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

Some of Euclid's other writings 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 figures 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 (mathematical sophisms);

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. with French N. Kurganova. SPb., 1769.288 p.
• Euclidean the elements of eight books, namely: 1st, 2nd, 3rd, 4th, 5th, 6th, 11th and 12th. / Per. from Greek. SPb.,. 370 pp.
  • 2nd ed. ... books 13 and 14 are attached to this sim. 1789.424 pp.
• Euclidean beginnings eight books, namely: the first six, 11th and 12th, containing the foundations of geometry. / Per. F. Petrushevsky. SPb., 1819.480 p.
• Euclidean began three books, namely: 7th, 8th and 9th, containing the general theory of numbers of ancient geometers. / Per. F. Petrushevsky. SPb., 1835.160 p.
• Eight books of geometry Euclid... / Per. with him. pupils of a real school ... Kremenchug, 1877. 172 pp.
• Beginnings Euclid... / C int. and interpretations

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

The main work of Euclid is called the Beginnings. Books with the same title, which consistently set out all the basic facts of geometry and theoretical arithmetic, were previously compiled by Hippocrates of Chios, Leont 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 proved 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 "Catoptrika", 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 presentation in all these works, as in the "Principles", 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 expression, which has become a winged expression, has come down to us.

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, a botanical and a zoological garden, an astronomical study, 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 English translation, 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's biography is very interesting for both adults and schoolchildren. He is the greatest ancient Greek philosopher, mathematician, optician, astronomer and musician of Hellenistic Egypt.

Who is it, who was it and why is it known? What is his contribution to mathematics, what is known from his biography, what is his social portrait? We will briefly talk about this and many other things below.

short biography

The biography of Euclid is not fully understood, for example, the year of birth is still unknown. It is known that he was born in a small area of ​​Athens and was a Platonic student.

The rise of his scientific work fell on the reign of Ptolemy the First. Some information about his life can be traced from Arabic manuscripts and Archimedean letters to friends. So, according to them, it can be determined that Euclid was the son of a Greek scientist and lived near Tire in Syria.

From an early age he received knowledge about the world from his father, he also instilled in his son a love of natural sciences, and then Euclid entered the school of Plato, where he studied mathematical foundations.

Having matured, he was invited to the Museion temple (according to other sources, he was one of its founders), in which prominent scientists and poets gathered. There were classes for classes. The temple was also filled with gardens with astronomy towers, lonely contemplation rooms and a large library.

In Museion, he was able to open a school with the best mathematicians and a monumental work in the field of mathematics, in which he laid the planimetric foundations with stereometry, number theory, the laws of algebra, methods for finding areas with volumes, etc.

Fragment of papyrus with the text "Beginnings" of Euclid

A monumental work - the publication of The Beginning. It is a 13-book series representing edited publications of ancient Greek mathematicians from the fifth to fourth centuries BC.

In addition to the "Elements", another essay was created - "Data", in which the foundations of geometric analysis were published. In addition, the Alexandrian scientist created a textbook, with the help of which astronomy, perspective, reflection in a mirror, musical intervals and trigonometric problems are studied at that time and now.

He devoted all the remaining years of his life to the study of natural sciences and mathematical laws, which is why he is called the father of geometry. Other aspects of his life are still unknown. He died in Alexandria.

Scientific activity and discoveries

The entire life of the scientist passed within the walls of Alexandria, therefore, his scientific activity with discoveries took place here. He received his education from the Platonic students, therefore he adopted the views from them, which helped him to form his mathematics class and become a teacher.

Euclid's predecessors were the famous mathematicians Thales with Pythagoras and Aristotle, who made fundamental discoveries in the field of trigonometric science. But these were scattered parts and did not represent one large logical chain.

Like his contemporaries, the mathematician and his students loved systematic and logical knowledge. That is why Euclid threw all his scientific activity on the systematization of previously obtained knowledge and their addition. In each of his books "Elements", he gives the basic concepts used by scientists earlier, and then introduces the basic axioms and postulates of geometry, which simplified the work of his descendants.

So, from the first to the fourth book, concepts and postulates from the works of Pythagoras and his followers are given, in the fifth book - the doctrine of proportions, from the sixth to the ninth book - knowledge about numbers, and in the last - publications on areas with planes and spaces (basics of stereometry ), irrationality, the doctrine of correct bodies.

The scientist made his discoveries in the same area. He introduced the concept of a point, line, plane and motion, developed postulates for the creation of certain geometric shapes in any area, the concept of light, mirrors, refraction of light rays, introduced the elementary theory of music, created a work on the use of geometry in the study of astronomy and errors that arise when forming geometric proofs.

In addition, the mathematician made small discoveries in the field of mechanics and gave the concept of the specific gravity of bodies.

Euclid is the father of geometry

It is not for nothing that Euclid is considered the father of geometry, since it was he who systematized the early knowledge gained from other famous mathematicians and philosophers of the past and provided the basis for the subsequent study of mathematics. He showed how flat surfaces and 3D geometry work.

Studying mathematics on a par with the followers of Plato, he ordered laws, spheres with cones and other geometric shapes. Hence the concept of Euclidean mathematician or Euclidean geometry is known.

It is he who owns the foundation of the principles in the form of axioms, which are taught today in all educational institutions. Thanks to Euclid, the principle of the plane of things and their measurability appeared, the idea of ​​13 elements that emphasize the importance of geometry and their use in everyday life.

Euclid was the first to simplify knowledge with the help of the books he wrote. He was the first to put geometry in a logical framework and make it easier to research. His ideas were able to shed light on the use of geometric data in life, to solve related problems and the use of conical sections to reveal the great perspectives of curves with cones being part of geometry.

Euclid's main work

The main work of the scientist is the written monument "Beginning". This is a book written about 300 BC and devoted to the systematic form of constructions in geometry.

This is the pinnacle of ancient geometry with ancient mathematics in general, which allowed further research and discoveries in the field of mathematics. The collection "Beginnings" is on a par with the work of Autolycus in importance.

It is interesting that the scientist's works are known only by mentions. The treatise "Beginnings" greatly influenced the mathematical development. The book was translated into hundreds of world languages ​​and is still used in teaching. In its importance at the time of publication, it was on a par with the Bible.

Euclid's achievements

Euclid's achievements were of great importance for world history, mathematics and other sciences.

He was the first to:

• systematized the well-known works of predecessors into a single collection of 13 books;
• created 5 postulates of GCD and 5 axioms in the field of geometry;
• characterized all known geometric figures, gave the concept of curved lines, conical sections and other phenomena;
• created a treatise on errors in the study and creation of geometric proofs;
• proved the practical use of mathematics in the study of stars, celestial bodies, space and other sciences;
• studied light with the laws of its propagation;
• studied mirrors and the ability to refract light rays in them;
• created the simplest theory in the field of music;
• created postulates and formulas for mechanics and determined the specific gravity of bodies.

Maths

Euclid is the father of mathematics. He formulated theorems on planimetry, simplified the understanding of the Pythagorean theorem and the theorem on the sum of the angles of a triangle, prescribed the properties of regular polygons and the laws for constructing regular fifteenagons, indicated how algebra is applicable in life and what its basic theories are, wrote down the theory of the whole and the rational number, considered the quadratic irrationality, laid the foundations of stereometric science, proved theorems concerning the area of ​​a circle with the volume of a sphere, derived the ratio of the volume of pyramids with cones, prisms and cylinders.

Other sciences

In addition to mathematics, the scientist worked with optics, astronomy, logic and music. So, in optics, he gave information about optical perspective, specular distortions and reflections of light rays in a mirror.

A few interesting facts from the biography of Euclid:

1. The oldest known mathematical treatise belongs to Euclid.
2. There is still no data on the place of birth and death of the great scientist. However, the place of Euclid's occupation is known about 2400 years ago and its location is Alexandria. Interestingly, this town today is the second largest in Egypt after Cairo;
3. Euclid was able to create 4 books on the conical section.
4. The fundamental work "Beginning" is considered so important for science that it is still used in life. It is interesting that there are other publications with a similar title, but the most popular is the work of Euclid. "
5. From his very youth, Euclid studied with the eminent scientist Plato, who taught Aristotle in Ancient Greece. Plato himself studied with Socrates.
6. Traditionally, geometry today bears the name of this scientist.
7. There is a legend that when once a student of the greatest mathematician asked him how geometry could help him in life, Euclid gave him money and kicked him out of class.
8. Euclid is still considered the author of numerous books, whose authorship has not been confirmed. These are various works, for example, publications on music, philosophy and medicine. It is officially known that the great scientist made a discovery in the optical and astronomical fields.
9. Today Riemannian, Lobachevian and Euclidean geometry is recognized. The latter is the most traditional and frequently used one.
10. The first time a Euclidean work was translated was at the end of the eighteenth century. At the same time, "Elements" were first translated into Armenian in the eleventh century.
11. Favorite phrase: "There is no royal way in geometry."

In general, Euclid is the father of geometry, and it is not by chance that he is called that. He was the first to make the complex understandable and gave impetus to the development of natural sciences. His books are invaluable in their value and are applied today in the field of mathematical and geometric sciences all over the world.