Ancient Greek mathematician Euclid: biography of the scientist, discoveries and interesting facts. Euclid and his contribution to geometry Basic provisions of the Elements

Euclid was born around 330 BC, presumably in Alexandria. Some Arab authors believe that he came from a wealthy family from Nocrates. There is a version that Euclid could have been born in Tyre, and spent his entire future life in Damascus. According to some documents, Euclid studied at the ancient school of Plato in Athens, which was only possible for 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 locate any reliable sources for the mathematician's biography. 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, making revolutionary discoveries in this area and proving many theorems. At that time, Alexandria was not only the 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 raised it to the 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 scholars of that time, wisely spent most of his time in the Library of Alexandria. This museum, dedicated to literature, art and sciences, was founded by Ptolemy. Here Euclid begins to unite geometric principles, arithmetic theories and irrational numbers into a single science, geometry. He continues to prove his theorems and compiles them into the colossal work “Principia.”

Over the entire period of his little-researched scientific activity, the scientist completed 13 editions of “Principles”, covering a wide range of issues, starting with axioms and statements and ending with stereometry and the theory of algorithms. Along with putting forward various theories, he begins to develop methods of proof and logical justification for these ideas, which 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 of a 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 in their work by various mathematicians and authors of 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 basic arithmetic knowledge. Through a chain of logical reasoning, Euclid sought to reveal 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 in completely different areas, including “arithmetic and numbers.” He concluded that discovering the "largest prime number" was 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 thought, 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 concepts such as “the whole is greater than the sum of the 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 Principia was translated from Greek into Arabic and then into English by John Dee. Since its writing, Principia has been reprinted 1,000 times and eventually found a place of honor in 20th-century classrooms. There are many cases where mathematicians tried to challenge and refute the geometric and mathematical theories of 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. It was only a century later that a new group of mathematicians would 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 a new field of optics - catoptrics, which largely established the mathematical function of 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 opened a private school at the Library of Alexandria in order to be able to teach mathematics to enthusiasts like himself. There is also an opinion that in the later period of 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 the scientist, mathematicians from 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”.

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

Euclid is the first mathematician of the Alexandrian school. His main job "Beginnings"(Στοιχεῖα, in Latinized form - “Elements”) contains a presentation of planimetry, stereometry and a number of issues in number theory; in it he summed up the previous development of Ancient Greek mathematics and created the foundation for the further development of mathematics.

Among other works on mathematics it should be noted "On the division of figures", preserved in Arabic translation, 4 books “Conic 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” of Pappus of Alexandria. Euclid - author of works on astronomy, optics, music, etc.

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

Additional touches to the portrait of Euclid can be gleaned from Pappus and Stobaeus. Pappus reports that Euclid was gentle and kind to everyone who could contribute even in the slightest degree to the development of mathematical sciences, and Stobaeus relates another anecdote about Euclid.

Having begun to study geometry and having analyzed the first theorem, one young man asked Euclid: “What benefit will I get from this science?” Euclid called the slave and said: “Give him three obols, since he wants to make a profit from his studies.” The historicity of the story is questionable, since a similar one is told about Plato.

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

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

Euclid's "Elements":

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

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

In Book I the properties of triangles and parallelograms are studied; This book is crowned with the famous theorem for right triangles.

Book II, going back to the Pythagoreans, is devoted to the so-called “geometric algebra.”

Books III and IV describe the geometry of circles, as well as inscribed and circumscribed polygons; when working on these books, Euclid could have used the writings of Hippocrates of Chios.

In Book V, the general theory of proportions constructed by Eudoxus of Cnidus is introduced, and in Book VI it is applied to the theory of similar figures.

Books VII-IX are devoted to number theory and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. These books discuss theorems on proportions and geometric progressions, introduce a method for finding the greatest common divisor of two numbers (now known as the Euclid algorithm), construct even perfect numbers, and prove the infinity of the set of prime numbers.

In Book X, which represents the most voluminous and complex part of the Elements, a classification of irrationalities is constructed; it is possible that its author is Theaetetus of Athens.

Book XI contains the basics of stereometry.

In the XII book, using the method of exhaustion, theorems on the ratios of the areas of circles, as well as the volumes of pyramids and cones are proved; The author of this book is generally acknowledged to be Eudoxus of Cnidus.

Finally, Book XIII is devoted to the construction of five regular polyhedra; It is believed that some of the constructions were developed by Theaetetus of Athens.

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

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

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

Biography

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

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

Some modern authors interpret Proclus's statement - Euclid lived in the time of Ptolemy I Soter - to mean that Euclid lived at the court of Ptolemy and was the founder of the Alexandrian Museion. It should be noted, however, that this idea was established in Europe in the 17th century, while medieval authors identified Euclid with the student of Socrates, the philosopher Euclid of Megara. An anonymous 12th-century Arabic manuscript reports:

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

According to his philosophical views, Euclid was most likely a Platonist.

Beginnings Euclid

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

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

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

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

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

Other works of Euclid

Statue of Euclid at the Oxford University Museum of Natural History

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

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

From brief descriptions we know:

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

Euclid is also credited with:

Euclid and ancient philosophy

The Greek treatise of Pseudo-Euclid with Russian translation and notes by G. A. Ivanov was published in Moscow in 1894

Literature

Bibliography

Max Stack. Bibliographia Euclideana. Die Geisteslinien der Tradition in den Editionen der “Elemente” des Euklid (um 365-300). Handschriften, Inkunabeln, Frühdrucke (16.Jahrhundert). Textkritische Editionen des 17.-20. Jahrhunderts. Editionen der Opera minora (16.-20.Jahrhundert). Nachdruck, herausgeg. von Menso Folkerts. Hildesheim: Gerstenberg, 1981.

Texts and translations

Old Russian translations

Euclidean elements from twelve non-phthonic books were selected and reduced into eight books through the professor of mathematics A. Farkhvarson. / Per. from lat. I. Satarova. St. Petersburg, 1739. 284 pp.
Elements of geometry, that is, the first foundations of the science of measuring distance, consisting of axis Euclidean books. / Per. from French N. Kurganova. St. Petersburg, 1769. 288 pp.
Euclidean elements eight books, namely: 1st, 2nd, 3rd, 4th, 5th, 6th, 11th and 12th. / Per. from Greek St. Petersburg, . 370 pp.
- 2nd ed. ...books 13 and 14 are attached to this. 1789. 424 pp.
Euclidean principles eight books, namely: the first six, 11th and 12th, containing the foundations of geometry. / Per. F. Petrushevsky. St. Petersburg, 1819. 480 pp.
Euclidean began three books, namely the 7th, 8th and 9th, containing the general theory of numbers of ancient geometers. / Per. F. Petrushevsky. St. Petersburg, 1835. 160 pp.
Eight books of geometry Euclid. / Per. with him. pupils of a real school... Kremenchug, 1877. 172 pp.
Beginnings Euclid. / From input. and interpretations by M.E. Vashchenko-Zakharchenko. Kyiv, 1880. XVI, 749 pp.

Modern editions of Euclid's works

The beginnings of Euclid. Per. and comm. D. D. Mordukhai-Boltovsky, ed. with the participation of I. N. Veselovsky and M. Ya. Vygodsky. In 3 volumes (Series “Classics of Natural History”). M.: GTTI, 1948-50. 6000 copies

Books I-VI (1948. 456 pp.) on www.math.ru or on mccme.ru
Books VII-X (1949. 512 pp.) on www.math.ru or on mccme.ru
Books XI-XIV (1950. 332 pp.) on www.math.ru or on mccme.ru

Euclidus Opera Omnia. Ed. I. L. Heiberg & H. Menge. 9 vols. Leipzig: Teubner, 1883-1916.

Vol. I-IX at www.wilbourhall.org

Heath T. L. The thirteen books of Euclid's Elements. 3 vols. Cambridge UP, 1925. Editions and translations: Greek (ed. J. L. Heiberg), English (ed. Th. L. Heath)
Euclide. Les elements. 4 vols. Trad. et comm. B. Vitrac; intr. M. Caving. P.: Presses universitaires de France, 1990-2001.
Barbera A. The Euclidian Division of the Canon: Greek and Latin Sources // Greek and Latin Music Theory. Vol. 8. Lincoln: University of Nebraska Press, 1991.

Comments

Antique comments Began

Proclus Diadochos. Commentaries on the first book of Euclid's Elements. Introduction. Per. and comm. Yu. A. Shichalina. M.: GLK, 1994.
Proclus Diadochos. Commentaries on the first book of Euclid's Elements. Postulates and axioms. Per. A. I. Shchetnikova. ΣΧΟΛΗ , vol. 2, 2008, p. 265-276.
Proclus Diadochos. Commentary on the first book of Euclid's Elements. Definitions. Per. A. I. Shchetnikova. Arche: Proceedings of the cultural-logical seminar, vol. 5. M.: RSUH, 2009, p. 261-320.
Thompson W. Pappus’ commentary on Euclid’s Elements. Cambridge, 1930.

Research

ABOUT Beginnings Euclid

Alimov N. G. Magnitude and relation in Euclid. Historical and mathematical research, vol. 8, 1955, p. 573-619.
Bashmakova I. G. Arithmetic books of Euclid’s Elements. , vol. 1, 1948, p. 296-328.
Van der Waerden B. L. Waking Science. M.: Fizmatgiz, 1959.
Vygodsky M. Ya. “Principles” of Euclid. Historical and mathematical research, vol. 1, 1948, p. 217-295.
Glebkin V.V. Science in the context of culture: (“Euclides’ Elements” and “Jiu Zhang Xuan Shu”). M.: Interprax, 1994. 188 pp. 3000 copies. ISBN 5-85235-097-4
Kagan V.F. Euclid, his successors and commentators. In the book: Kagan V.F. Foundations of Geometry. Part 1. M., 1949, p. 28-110.
Raik A.E. The tenth book of Euclid’s Elements. Historical and mathematical research, vol. 1, 1948, p. 343-384.
Rodin A.V. Mathematics of Euclid in the light of the philosophy of Plato and Aristotle. M.: Nauka, 2003.
Tseyten G. G. History of mathematics in antiquity and the Middle Ages. M.-L.: ONTI, 1938.
Shchetnikov A.I. The second book of Euclid’s “Principles”: its mathematical content and structure. Historical and mathematical research, vol. 12(47), 2007, p. 166-187.
Shchetnikov A.I. The works of Plato and Aristotle as evidence of the formation of a system of mathematical definitions and axioms. ΣΧΟΛΗ , vol. 1, 2007, p. 172-194.

Artmann B. Euclid’s “Elements” and its prehistory. Apeiron, v. 24, 1991, p. 1-47.
Brooker M.I.H., Connors J.R., Slee A.V. Euclid. CD-ROM. Melbourne, CSIRO-Publ., 1997.
Burton H.E. The optics of Euclid. J. Opt. Soc. Amer., v. 35, 1945, p. 357-372.
Itard J. Lex livres arithmetiqués d'Euclide. P.: Hermann, 1961.
Fowler D.H. An invitation to read Book X of Euclid’s Elements. Historia Mathematica, v. 19, 1992, p. 233-265.
Knorr W.R. The evolution of the Euclidean Elements. Dordrecht: Reidel, 1975.
Mueller I. Philosophy of mathematics and deductive structure in Euclid’s Elements. Cambridge (Mass.), MIT Press, 1981.
Schreiber P. Euclid. Leipzig: Teubner, 1987.
Seidenberg A. Did Euclid’s Elements, Book I, develop geometry axiomatically? Archive for History of Exact Sciences, v. 14, 1975, p. 263-295.
Staal J.F. Euclid and Panini // Philosophy East and West. 1965. No. 15. P. 99-115.
Taisbak C.M. Division and logos. A theory of equivalent couples and sets of integers, propounded by Euclid in the arithmetical books of the Elements. Odense UP, 1982.
Taisbak C.M. Colored quadrangles. A guide to the tenth book of Euclid's Elements. Copenhagen, Museum Tusculanum Press, 1982.
Tannery P. La geometrié grecque. Paris: Gauthier-Villars, 1887.

About other works of Euclid

Zverkina G. A. Review of Euclid’s treatise “Data”. Mathematics and practice, mathematics and culture. M., 2000, p. 174-192.
Ilyina E. A. About the “Data” of Euclid. Historical and mathematical research, vol. 7(42), 2002, p. 201-208.
Shawl M. // . M., 1883.
Berggren J.L., Thomas R.S.D. Euclid's Phaenomena: a translation and study of a Hellenistic treatise in spherical astronomy. NY, Garland, 1996.
Schmidt R. Euclid's Recipients, commonly called the Data. Golden Hind Press, 1988.
S. Kutateladze Apology of Euclid

Notes

Links

Euclid is the first mathematician of the Alexandrian school. His main work “Principia” (????????, in Latinized form - “Elements”) contains a presentation of planimetry, stereometry and a number of questions in number theory; in it he summed up the previous development of Greek mathematics and created the foundation for the further development of mathematics. Among other works on mathematics, it should be noted “On the division of figures”, preserved in Arabic translation, 4 books “Conic Sections”, the material of which was included in the work of the same title by Apollonius of Perga, as well as “Porisms”, an idea of which can be obtained from the “Mathematical collection" by Pope of Alexandria. Euclid - author of works on astronomy, optics, music, etc.

Biography

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

According to his philosophical views, Euclid was most likely a Platonist.

Euclid's Elements

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

Other works of Euclid

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

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

From brief descriptions we know:

Porisms (?????????) - about the conditions that determine curves;
Conic sections (??????);
Superficial places (????? ???? ?????????) - about the properties of conic sections;
Pseudarius (??????????) - about errors in geometric proofs;

Euclid is also credited with:

Catoptrics (????????????) - theory of mirrors; the treatment of Theon of Alexandria has survived;
Division of the Canon (???????? ?????????) - a treatise on elementary music theory.

Euclid and ancient philosophy

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

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

Some “Platonism” of Euclid is also connected with the fact that in Plato’s Timaeus the doctrine of the four elements is considered, which correspond to four regular polyhedra (tetrahedron - fire, octahedron - air, icosahedron - water, cube - earth), the fifth polyhedron, dodecahedron, “ belonged to the figure of the universe." In this regard, the Principia can be considered as a doctrine developed with all the necessary premises and connections about the construction of five regular polyhedra - the so-called “Platonic solids”, culminating in the proof of the fact that there are no other regular solids besides these five.

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

Pseudo-Euclid

Euclid is credited with two important treatises on ancient music theory: the Harmonic Introduction and the Division of the Canon. Nothing is known about the real author of these works. Henry Meibom (1555-1625) provided the Harmonic Introduction with extensive notes, and, together with the Division of the Canon, was the first to authoritatively attribute them to the works of Euclid. With the subsequent detailed analysis of these treatises, it was determined that the first has traces of the Pythagorean tradition (for example, in it all semitones are considered equal), and the second is distinguished by an Aristotelian character (for example, the possibility of dividing a tone in half is denied). The style of presentation of the “Harmonic Introduction” is distinguished by dogmatism and continuity; the style of the “Division of the Canon” is somewhat similar to Euclid’s “Elements”, since it also contains theorems and proofs.

Karl Jahn (1836-1899) was of the opinion that the treatise “Harmonic Introduction” was written by Kleonidas, since his name appears in some manuscripts. In addition to the names of Euclid and Cleonidas, the manuscripts mention Pappus and Anonymous as authors. In most scientific publications, they prefer to call the author Pseudo-Euclid.

The Greek treatise of Pseudo-Euclid with Russian translation and notes by G. A. Ivanov was published in Moscow in 1894

Euclid (365-300 BC), ancient Greek mathematician.

Born in Athens (according to other sources, in Tire). All that is known for sure about the scientist’s life is that he was a student of Plato, and the heyday of his activity occurred during the reign of Ptolemy I Soter in Egypt (IV century BC).

The name of Euclid is mentioned in a letter from Archimedes to friends, for example to the philosopher Dositheus (“On the Ball and the Cylinder”). Some biographical data has been preserved on the pages of an Arabic manuscript of the 12th century: “Euclid, son of Naukrates, known as Geometra, a scientist of old times, Greek by origin, Syrian by residence, originally from Tyre.”

During the time of Ptolemy, Alexandria, the capital of the Egyptian kingdom, was a major cultural center. In order to exalt his state, Ptolemy summoned scientists and poets to the country, creating for them a temple of muses - the Museion. There were study rooms, botanical and zoological gardens, an astronomical tower, rooms for solitary work, and most importantly, the magnificent Library of Alexandria.

Among the invitees was Euclid, who founded a mathematical school here and created for his students a fundamental work on geometry under the general title “Elements” (about 325 BC). It outlines the basics of planimetry, stereometry, number theory, algebra, describes methods for determining areas and volumes, etc.

"Principles" consists of 15 books. In part, they represent an adaptation of treatises by Greek mathematicians of the V-IV centuries. BC e. No scientific book has ever enjoyed such popularity; it was even said that after the Bible it was the most popular written monument of antiquity. The Elements were copied on papyrus; parchment, paper, and then by printing (for the first time in 1533 in Basel, Switzerland). Up to the 20th century. the book was considered a basic textbook on geometry not only for schools, but also for universities.

Another significant work of Euclid, “Data,” is an introduction to geometric analysis. The scientist also owns “Phenomena” (dedicated to elementary spherical astronomy), “Optics” (contains the doctrine of perspective) and “Catoptrics” (explains the theory of reflections in mirrors), a small treatise “Sections of the Canon” (includes ten problems on musical intervals), a collection problems on dividing the areas of figures “On divisions” (came down to us in Arabic translation).

Euclid presumably died in Alexandria.