To the most reliable information about the life of Euclid, it is customary to attribute the little that is given in the Commentaries of Proclus to the first book Began Euclid. Noting that “the mathematicians who wrote on the history” did not bring the development of this science to the time of Euclid, Proclus points out that Euclid was older than the Platonic 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 the portrait of Euclid can be gleaned from Pappus and Stobeus. Papp reports that Euclid was gentle and amiable with everyone who could contribute even in the slightest degree to the development of mathematical sciences, and Stobaeus relates another anecdote about Euclid. Having begun the study of geometry and having analyzed the first theorem, one young man asked Euclid: “And what will be the benefit to me 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 doubtful, as a similar story is told about Plato.

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

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

Euclid, son of Naucrates, known under the name of "Geometer", a scientist of the old time, Greek by origin, Syrian by residence, originally from Tyre ...

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

« Beginnings» Euclid

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

Beginnings consists 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 line can be drawn through any two points"), and axioms - general rules output when operating with values ​​(for example, “if two values ​​\u200b\u200bare equal to a third, they are equal to each other”).

Book I studies the properties of triangles and parallelograms; this book is crowned by the famous Pythagorean theorem for right triangles. Book II, dating back to the Pythagoreans, is devoted to the so-called "geometric algebra". Books III and IV deal with the geometry of circles, as well as inscribed and circumscribed polygons; when working on these books, Euclid could use the writings of Hippocrates of Chios. Book V introduces the general theory of proportions built by Eudoxus of Cnidus, and in Book VI it is applied to the theory of similar figures. Books VII-IX are devoted to the theory of numbers and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. These books deal with theorems on proportions and geometric progressions, introduce a method for finding the greatest common divisor of two numbers (now known as Euclid's algorithm), construct even perfect numbers, and prove the infinity of the set of primes. 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 fundamentals of stereometry. In Book XII, using the exhaustion method, theorems are proved on 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 Theaetetus of Athens.

In the manuscripts that have come down to us, two more have been 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, the builder of the church of St. Sophia in Constantinople (beginning of the 6th century AD).

Beginnings provide a common basis for subsequent geometric treatises by Archimedes, Apollonius, and other ancient authors; the propositions proved in them are considered to be well known. Comments on Beginnings in antiquity they were Heron, Porphyry, Pappus, Proclus, Simplicius. A commentary by Proclus to Book I has been preserved, as well as a commentary by Pappus to 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 an example of a mathematical treatise, strictly and systematically expounding the main provisions of a particular mathematical science.

Other works by Euclid

From other writings of Euclid survived:

• Data (δεδομένα ) - about what is needed to set the figure;
• About division (περὶ διαιρέσεων ) - preserved partially and only in 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 the rectilinear propagation of light.

The short descriptions are:

• porisms (πορίσματα ) - about the conditions that determine the curves;
• Conic 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

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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 Nephtonian books selected and reduced to eight books through the professor of mathematics A. Farhvarson. / Per. from lat. I. Satarova. SPb., 1739. 284 pages.
• 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 pp.
• Euclidean Elements 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. 1789. 424 pages.
• Euclidean principles eight books, namely the first six, the 11th and the 12th, containing the foundations of geometry. / Per. F. Petrushevsky. SPb., 1819. 480 pages.
• Euclidean began three books, namely: 7th, 8th and 9th, containing the general theory of numbers of ancient geometers. / Per. F. Petrushevsky. SPb., 1835. 160 pages.
• Eight books of geometry Euclid. / Per. with him. pupils of a real school ... Kremenchug, 1877. 172 p.
• Beginnings Euclid. / From input. and interpretations of M. E. Vashchenko-Zakharchenko. Kyiv, 1880. XVI, 749 pages.

Modern editions of Euclid's writings

• Beginnings of Euclid. Per. and comm. D. D. Mordukhai-Boltovsky, ed. participation of I. N. Veselovsky and M. Ya. Vygodsky. In 3 volumes (Series "Classics of Natural Science"). M.: GTTI, 1948-50. 6000 copies

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

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

• Vol. I-IX on

• Heath T.L. The third books of Euclid's Elements. 3vols. Cambridge UP, 1925. Editions and translations: .
• Euclide. Les elements. 4 vols. Trad. et comm. B. Vitrac; intr. M. Caveing. P.: Presses universitaires de France, 1990-2001.
• Barber 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 Diadochus. . Per. and comm. Yu. A. Shichalina. M.: GLK, 1994.
• Proclus Diadoch. Commentary on the first book of Euclid's "Beginnings" / Translation by A. I. Shchetnikov. - M .: Russian Foundation for the Promotion of Education and Science, 2013.
• Thompson W. Pappus' commentary on Euclid's Elements. Cambridge, 1930.

Research

O Beginnings Euclid

• Alimov N. G. Value and relation in Euclid. Historical and mathematical research, issue. 8, 1955, p. 573-619.
• Bashmakova I. G. Arithmetic books of the "Beginnings" of Euclid. , issue. 1, 1948, p. 296-328.
• Van der Waerden B. L. Awakening Science. Moscow: Fizmatgiz, 1959.
• Vygodsky M. Ya. "Beginnings" of Euclid. Historical and mathematical research, issue. 1, 1948, p. 217-295.
• Glebkin V.V. Science in the context of culture: ("Beginnings" by Euclid and "Jiu zhang suan shu"). Moscow: Interpraks, 1994. 188 pages, 3000 copies. ISBN 5-85235-097-4
• Kagan VF 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 "Beginnings". Historical and mathematical research, issue. 1, 1948, p. 343-384.
• Rodin A.V. Euclid's mathematics in the light of the philosophy of Plato and Aristotle. M.: Nauka, 2003.
• Zeiten G. G. History of mathematics in antiquity and the Middle Ages. M.-L.: ONTI, 1938.
• Shchetnikov AI The second book of Euclid's "Beginnings": its mathematical content and structure. Historical and mathematical research, issue. 12(47), 2007, p. 166-187.
• Shchetnikov AI Works of Plato and Aristotle as evidence of the formation of a system of mathematical definitions and axioms. ΣΧΟΛΗ , issue. 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 arithmetiques 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. Eulid. 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 squares. A guide to the tenth book of Euclid's Elements. Copenhagen, Museum Tusculanum Press, 1982.
• Tannery P. La geometrie grecque. Paris: Gauthier-Villars, 1887.

On other writings 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, issue. 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

see also

Notes

Links

• Khramov Yu. A. Euclid // Physicists: Biographical Directory / Ed. A. I. Akhiezer. - Ed. 2nd, rev. and additional - M .: Science, 1983. - S. 109. - 400 p. - 200,000 copies.(in trans.)

The science Philosophy Philology Literature

An excerpt characterizing Euclid

“Oh, how heavy is this incessant nonsense!” thought Prince Andrei, trying to drive this face out of his imagination. But this face stood before him with the force of reality, and this face drew nearer. Prince Andrei wanted to return to the former world of pure thought, but he could not, and delirium drew him into his own realm. A quiet whispering voice continued its measured babble, something pressed, stretched, and a strange face stood before him. Prince Andrei gathered all his strength to come to his senses; he stirred, and suddenly there was a ringing in his ears, his eyes became dim, and he, like a man who has plunged into water, lost consciousness. When he woke up, Natasha, that very living Natasha, whom, of all the people in the world, he most of all wanted to love with that new, pure divine love that was now revealed to him, was kneeling before him. He realized that it was alive the real Natasha, and was not surprised, but quietly delighted. Natasha, on her knees, frightened, but chained (she could not move), looked at him, holding back her sobs. Her face was pale and motionless. Only in the lower part of it fluttered something.
Prince Andrei breathed a sigh of relief, smiled and held out his hand.
- You? - he said. - How happy!
Natasha with a quick but careful movement moved towards him on her knees and, carefully taking his hand, bent over her face and began to kiss her, slightly touching her lips.
- Sorry! she said in a whisper, raising her head and looking at him. - Forgive me!
“I love you,” said Prince Andrei.
- Sorry…
- Forgive what? asked Prince Andrew.
“Forgive me for what I did,” Natasha said in a barely audible, interrupted whisper and began to kiss her hand more often, slightly touching her lips.
“I love you more, better than before,” said Prince Andrei, raising her face with his hand so that he could look into her eyes.
Those eyes, filled with happy tears, looked at him timidly, compassionately and joyfully with love. Natasha's thin and pale face with swollen lips was more than ugly, it was terrible. But Prince Andrei did not see this face, he saw shining eyes that were beautiful. Behind them, a voice was heard.
Pyotr the valet, now completely awake from sleep, woke the doctor. Timokhin, who could not sleep all the time because of the pain in his leg, had long seen everything that was being done, and, diligently covering his undressed body with a sheet, huddled on the bench.
- What is this? said the doctor, rising from his bed. “Let me go, sir.”
At the same time, a girl knocked on the door, sent by the countess, missing her daughter.
Like a somnambulist who was awakened in the middle of her sleep, Natasha left the room and, returning to her hut, fell on her bed sobbing.

From that day on, during the entire further journey of the Rostovs, at all rests and overnight stays, Natasha did not leave the wounded Bolkonsky, and the doctor had to admit that he did not expect from the girl either such firmness or such skill in walking after the wounded.
No matter how terrible the idea seemed to the countess that Prince Andrei could (very likely, according to the doctor) die during the journey in the arms of her daughter, she could not resist Natasha. Although, as a result of the now established rapprochement between the wounded Prince Andrei and Natasha, it occurred to me that in the event of recovery, the former relations between the bride and groom would be resumed, no one, still less Natasha and Prince Andrei, spoke about this: the unresolved, hanging question of life or death was not only over Bolkonsky, but over Russia obscured all other assumptions.

Pierre woke up late on September 3rd. His head ached, the dress in which he slept without undressing weighed heavily on his body, and in his soul there was a vague consciousness of something shameful that had been committed the day before; it was shameful yesterday's conversation with Captain Rambal.
The clock showed eleven, but it seemed especially overcast outside. Pierre got up, rubbed his eyes, and, seeing a pistol with a carved stock, which Gerasim put back on the desk, Pierre remembered where he was and what was coming to him that very day.
“Am I too late? thought Pierre. “No, he will probably make his entry into Moscow no earlier than twelve.” Pierre did not allow himself to think about what lay ahead of him, but was in a hurry to act quickly.
Adjusting his dress, Pierre took a pistol in his hands and was about to go. But then for the first time the thought came to him about how, not in his hand, along the street to carry this weapon to him. Even under a wide caftan it was difficult to hide a large pistol. Neither behind the belt nor under the arm could it be placed inconspicuously. In addition, the pistol was unloaded, and Pierre did not have time to load it. “It doesn’t matter, the dagger,” Pierre said to himself, although more than once, discussing the fulfillment of his intention, he decided with himself that the main mistake of the student in 1809 was that he wanted to kill Napoleon with a dagger. But, as if Pierre’s main goal was not to fulfill his plan, but to show himself that he did not renounce his intention and was doing everything to fulfill it, Pierre hastily took what he had bought from the Sukharev Tower along with a pistol a blunt serrated dagger in a green scabbard and hid it under his waistcoat.
Belting his caftan and pulling on his hat, Pierre, trying not to make noise and not meet the captain, walked along the corridor and went out into the street.
That fire, which he had looked at with such indifference the previous evening, increased significantly during the night. Moscow was already burning from different sides. Burning at the same time Karetny Ryad, Zamoskvorechye, Gostiny Dvor, Povarskaya, barges on the Moskva River and a wood market near the Dorogomilovsky Bridge.
Pierre's path lay through lanes to Povarskaya and from there to the Arbat, to Nikola Yavlenny, in whose imagination he had long ago determined the place where his deed should be done. Most of the houses had locked gates and shutters. The streets and lanes were deserted. The air smelled of burning and smoke. From time to time there were Russians with uneasily timid faces and Frenchmen with a non-urban, camp look, walking along the middle of the streets. Both of them looked at Pierre with surprise. In addition to his great height and thickness, in addition to the strange gloomy concentrated and suffering expression of his face and whole figure, the Russians looked closely at Pierre, because they did not understand what class this person could belong to. The French followed him with surprise with their eyes, especially because Pierre, disgusted by all other Russians, who looked at the French with fear or curiosity, did not pay any attention to them. At the gates of a house, three Frenchmen, who were explaining something to the Russian people who did not understand them, stopped Pierre, asking if he knew French?
Pierre shook his head negatively and went on. In another alley, a sentry standing at a green box shouted at him, and Pierre only realized at the repeated menacing cry and the sound of a gun taken by the sentry in his hand that he had to go around the other side of the street. He did not hear or see anything around him. He, like something terrible and alien to him, with haste and horror carried his intention within himself, fearing - taught by the experience of last night - somehow lose it. But Pierre was not destined to convey his mood intact to the place where he was heading. In addition, even if he had not been hindered by anything on the way, his intention could not have been fulfilled already because Napoleon had traveled more than four hours ago from the Dorogomilovsky suburb through the Arbat to the Kremlin and was now sitting in the tsar’s office in the gloomiest mood. Kremlin Palace and gave detailed, detailed orders on the measures that should immediately have been taken to extinguish the fire, prevent looting and calm the inhabitants. But Pierre did not know this; he, completely absorbed in what was to come, was tormented, as people are tormented who stubbornly undertook an impossible deed - not because of difficulties, but because of the unusualness of the matter with their nature; he was tormented by the fear that he would weaken at the decisive moment and, as a result, lose respect for himself.
Although he did not see or hear anything around him, he knew the way by instinct and was not mistaken by the lanes that led him to Povarskaya.
As Pierre approached Povarskaya, the smoke grew stronger and stronger, it even became warm from the fire. From time to time, fiery tongues rose from behind the roofs of houses. More people met on the streets, and this people were more anxious. But Pierre, although he felt that something unusual was going on around him, did not realize that he was approaching the fire. Walking along a path that ran along a large undeveloped place, adjacent on one side to Povarskaya, on the other to the gardens of the house of Prince Gruzinsky, Pierre suddenly heard a desperate cry of a woman beside him. He stopped, as if awakening from a dream, and raised his head.
Away from the path, on dried dusty grass, a heap of household belongings were piled up: featherbeds, a samovar, images and chests. On the ground near the chests sat a middle-aged, thin woman, with long protruding upper teeth dressed in a black coat and cap. This woman, swaying and saying something, bursting into tears. Two girls, from ten to twelve years old, dressed in dirty short dresses and cloaks, with an expression of bewilderment on their pale, frightened faces, looked at their mother. A younger boy, about seven years old, in a coat and a huge cap that was not his own, was crying in the arms of the old nurse. A dirty, barefooted girl sat on a chest and, having loosened her whitish braid, tugged at her singed hair, sniffing at it. The husband, a short, round-shouldered man in a uniform, with wheel-shaped sideburns and smooth temples that could be seen from under a straight-on cap, with a motionless face, parted chests stacked one on top of the other, and pulled out some kind of robes from under them.
The woman almost threw herself at Pierre's feet when she saw him.
“Dear fathers, Orthodox Christians, save me, help me, my dear! .. someone help me,” she said through sobs. - A girl! .. Daughter! .. They left my younger daughter! .. Burned down! Oh oh oh! for that I lele you ... Oh oh oh!
“That’s enough, Marya Nikolaevna,” the husband turned to his wife in a low voice, apparently only to justify himself before a stranger. - The sister must have taken it away, otherwise where else to be? he added.
- An idol! The villain! the woman screamed angrily, suddenly stopping crying. “You don’t have a heart, you don’t feel sorry for your child. Another would have taken it out of the fire. And this is an idol, not a man, not a father. You are a noble person, - the woman turned to Pierre with a patter, sobbing. - It caught fire nearby, - it was thrown towards us. The girl screamed: it's on fire! Rushed to collect. In what they were, they jumped out in that ... That's what they captured ... God's blessing and a dowry bed, otherwise everything was gone. Grab the kids, no Katechki. Oh my God! Ltd! – and again she sobbed. - My dear child, it burned down! burned down!
- Yes, where, where did she stay? Pierre said. From the expression on his animated face, the woman realized that this man could help her.
- Father! Father! she screamed, grabbing his legs. “Benefactor, at least calm my heart ... Aniska, go, vile, see her off,” she shouted at the girl, angrily opening her mouth and showing her long teeth even more with this movement.
“See, see, I ... I ... I will do it,” Pierre said hastily in a breathless voice.
The dirty girl stepped out from behind the trunk, cleaned up her scythe, and, sighing, went forward with her blunt bare feet along the path. Pierre, as it were, suddenly woke up to life after a severe fainting spell. He raised his head higher, his eyes lit up with the brilliance of life, and he quickly followed the girl, overtook her and went out to Povarskaya. The whole street was covered with a cloud of black smoke. Tongues of flame escaped from this cloud in some places. People crowded in front of the fire in a large crowd. In the middle of the street stood a French general and said something to those around him. Pierre, accompanied by a girl, went up to the place where the general was standing; but the French soldiers stopped him.
- On ne passe pas, [They don't pass here,] - a voice shouted to him.
- Over here, uncle! - said the girl. - We will go through the alley, through the Nikulins.
Pierre turned back and walked, occasionally jumping up to keep up with her. The girl ran across the street, turned left into an alley and, after passing through three houses, turned right at the gate.
“Right here now,” said the girl, and, running through the yard, she opened the gate in the boarded fence and, stopping, pointed out to Pierre a small wooden outbuilding that burned brightly and hotly. One side of it collapsed, the other burned, and the flames brightly knocked out from under the openings of the windows and from under the roof.
When Pierre entered the gate, he was overwhelmed with heat, and he involuntarily stopped.
- Which, which is your house? - he asked.
– Oh oh oh! howled the girl, pointing to the outbuilding. - He was the most, she was our most Vater. Burnt, you are my treasure, Katechka, my beloved lady, oh oh! Aniska howled at the sight of the fire, feeling the need to show her feelings as well.
Pierre leaned towards the outbuilding, but the heat was so strong that he involuntarily described an arc around the outbuilding and found himself near big house, which was still burning only on one side from the roof and around which a crowd of Frenchmen swarmed. At first, Pierre did not understand what these Frenchmen were doing, dragging something; but, seeing in front of him a Frenchman who beat a peasant with a blunt cleaver, taking away his fox coat, Pierre vaguely realized that they were robbing here, but he had no time to dwell on this thought.
The sound of the crackling and rumble of collapsing walls and ceilings, the whistling and hissing of flames and the lively cries of the people, the sight of fluctuating, then frowning thick black, then soaring brightening clouds of smoke with glitters of sparks and somewhere solid, sheaf-like, red, sometimes scaly gold, moving along the walls of the flame , the feeling of heat and smoke and the speed of movement produced their usual exciting effect on Pierre from fires. This effect was especially strong on Pierre, because Pierre suddenly, at the sight of this fire, felt freed from the thoughts that weighed on him. He felt young, cheerful, agile and determined. He ran around the outbuilding from the side of the house and was about to run to that part of it that was still standing, when a cry of several voices was heard above his very head, followed by the crackling and ringing of something heavy that fell beside him.
Pierre looked around and saw Frenchmen in the windows of the house, throwing out a chest of drawers filled with some kind of metal things. The other French soldiers below approached the box.
- Eh bien, qu "est ce qu" il veut celui la, [What else does this need,] one of the French shouted at Pierre.
– Un enfant dans cette maison. N "avez vous pas vu un enfant? [A child in this house. Have you seen the child?] - said Pierre.
- Tiens, qu "est ce qu" il chante celui la? Va te promener, [What else does this one interpret? Go to hell,] - voices were heard, and one of the soldiers, apparently afraid that Pierre would not take it into his head to take away the silver and bronze that were in the box, menacingly approached him.
- Unenfant? shouted a Frenchman from above. - J "ai entendu piailler quelque chose au jardin. Peut etre c" est sou moutard au bonhomme. Faut etre humain, voyez vous… [Child? I heard something squeaking in the garden. Maybe it's his child. Well, it is necessary for humanity. We are all human…]
– Ou est il? Ouestil? [Where is he? Where is he?] asked Pierre.
- Parici! Parici! [Here, here!] - the Frenchman shouted to him from the window, pointing to the garden that was behind the house. - Attendez, je vais descendre. [Wait, I'll get off now.]
And indeed, a minute later a Frenchman, a black-eyed fellow with some kind of spot on his cheek, in one shirt jumped out of the window of the lower floor and, slapping Pierre on the shoulder, ran with him into the garden.
“Depechez vous, vous autres,” he called to his comrades, “start a faire chaud.” [Hey, you, come on, it's starting to bake.]
Running outside the house onto a sandy path, the Frenchman pulled Pierre's hand and pointed him to the circle. Under the bench lay a three-year-old girl in a pink dress.
- Voila votre moutard. Ah, une petite, tant mieux, said the Frenchman. – Au revoir, mon gros. Faut etre humane. Nous sommes tous mortels, voyez vous, [Here is your child. Oh girl, so much the better. Goodbye, fat man. Well, it is necessary for humanity. All people,] - and the Frenchman with a spot on his cheek ran back to his comrades.
Pierre, choking with joy, ran up to the girl and wanted to take her in his arms. But, seeing a stranger, the scrofulous, mother-like, unpleasant-looking girl screamed and rushed to run. Pierre, however, grabbed her and lifted her up; she squealed in a desperately angry voice and with her small hands began to tear off Pierre's hands from herself and bite them with a snotty mouth. Pierre was seized by a feeling of horror and disgust, similar to that which he experienced when he touched some small animal. But he made an effort on himself not to abandon the child, and ran with him back to big house. But it was no longer possible to go back the same way; the girl Aniska was no longer there, and Pierre, with a feeling of pity and disgust, clutching the sobbing and wet girl as tenderly as possible, ran through the garden to look for another way out.

When Pierre, having run around the yards and lanes, went back with his burden to the Gruzinsky garden, at the corner of Povarskaya, for the first minute he did not recognize the place from which he went after the child: it was so cluttered with people and belongings pulled out of the houses. In addition to Russian families with their belongings, who were fleeing the fire here, there were also several French soldiers in various attire. Pierre ignored them. He was in a hurry to find the official's family in order to give his daughter to his mother and go again to save someone else. It seemed to Pierre that he still had a lot to do and that he needed to do it as soon as possible. Inflamed with heat and running around, Pierre at that moment, even stronger than before, experienced that feeling of youth, revival and determination that seized him while he ran to save the child. The girl was quiet now and, holding on to Pierre's caftan with her hands, sat on his arm and, like wild animal, looked around. Pierre glanced at her from time to time and smiled slightly. It seemed to him that he saw something touchingly innocent and angelic in that frightened and sickly little face.
In the same place, neither the official nor his wife was gone. Pierre walked with quick steps among the people, looking at the different faces that came across to him. Involuntarily, he noticed a Georgian or Armenian family, consisting of a very old man, handsome, with an oriental type of face, dressed in a new indoor sheepskin coat and new boots, an old woman of the same type and a young woman. This very young woman seemed to Pierre the perfection of oriental beauty, with her sharp, arched black eyebrows and a long, unusually tenderly ruddy and beautiful face without any expression. Among the scattered belongings, in the crowd in the square, she, in her rich satin coat and bright purple shawl that covered her head, resembled a tender hothouse plant thrown into the snow. She was sitting on knots a little behind the old woman and motionlessly with large black oblong eyes with long eyelashes looked at the ground. Apparently, she knew her beauty and was afraid for her. This face struck Pierre, and in his haste, passing along the fence, he looked back at her several times. Having reached the fence and still not finding those whom he needed, Pierre stopped, looking around.
The figure of Pierre with a child in her arms was now even more remarkable than before, and several people of Russian men and women gathered around him.
“Or did you lose someone, dear man?” Are you one of the nobles yourself? Whose child is that? they asked him.
Pierre answered that the child belonged to a woman and a black coat, who sat with the children in this place, and asked if anyone knew her and where she had gone.
“After all, it must be the Anferovs,” said the old deacon, turning to the pockmarked woman. “Lord have mercy, Lord have mercy,” he added in his usual bass.

Euclid was born around 330 BC, presumably in the city of Alexandria. Some Arabic 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 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 that, he moved to the city of 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 that of Euclid of Meguro, making it difficult to find any reliable source for the mathematician's life. It is only known for certain that it was he who attracted public attention to mathematics and brought this science to a completely new level, having made revolutionary discoveries in this area and proving many theorems. In those days, Alexandria was not only the largest city in the western part of the world, but also the center of a large, flourishing 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 sections of mathematics at that time. After moving to Alexandria, Euclid, like many scholars of the time, wisely spends most of his time in the Library of Alexandria. This museum, dedicated to literature, arts and sciences, was founded by Ptolemy. Here Euclid begins to combine geometric principles, arithmetic theories and irrational numbers into a single science of geometry. He continues to prove his theorems and reduces them to the colossal work of the Elements.

For all the time of his little-studied scientific activity, the scientist completed 13 editions of the "Beginnings", covering a wide range of issues, from axioms and statements to stereometry and the theory of algorithms. Along with putting forward various theories, he begins to develop a method of proof and a rationale 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 representations. It is known for certain that as one of the examples in his "Principles" Euclid used the Pythagorean theorem, which establishes the ratio 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 the "Beginnings", up to the 20th century, more copies of this book were sold than the Bible. The Elements, published and republished countless times, were used in their work by various mathematicians and authors of scientific papers. 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. By a chain of logical reasoning, Euclid sought to reveal secret knowledge to mankind. The system that the scientist continued to develop in his "Principles" will become the only geometry that the world will 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 the indisputable facts of theories. At a time when access to knowledge was limited, Euclid took up the study of issues in completely different areas, including "arithmetic and numbers." He concluded that finding the "largest prime number" is physically impossible. He substantiated 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 cannot be used in the construction of mathematical theories and formulas. He realized that even axioms had to be backed up irrefutable evidence. Therefore, the scientist began to give logical conclusions that confirmed his geometric axioms and theorems. For a better understanding of these axioms, he divided them into two groups, which he called "postulates". The first group is known as "general concepts", consisting of recognized scientific statements. The second group of postulates is synonymous with geometry itself. The first group includes such concepts 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 refer directly to geometry, stating that "all right angles are equal to each other", and that "a line can be drawn from any point to any point."

The scientific activity of the mathematician Euclid flourished, and in the early 1570s. his Elements were translated from Greek into Arabic and then into English by John Dee. Since its inception, The Elements has been reprinted 1,000 times and eventually won a place of honor in the classrooms of the 20th century. There are many cases when 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. And only a century later, a new group of mathematicians will be able to present innovative theories in the field of geometry.

Other jobs

Without ceasing to work on changing the theory of mathematics, Euclid managed to write a number of works on other topics that are used and referred to to this day. These writings were pure speculation based on irrefutable evidence that runs like a red thread through all of the "Beginnings". The scientist continued his study and discovered a new field of optics - catoptrics, which to a large extent approved the mathematical function of mirrors. His work in the field of optics, mathematical relations, systematization of data and the study of conic sections was lost in the mists of time. Euclid is known to have successfully completed eight editions, or books, on theorems concerning conic sections, but none of them have 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 "Beginnings". 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 in developing their own theories and writing works. We do not even have a clear idea of ​​​​the appearance of the scientist, 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 mankind. There are vague hints in the literature that he may have died around 260 BC. The legacy left by the scientist after himself is much more significant than the impression he made during his lifetime. His books and writings were sold all over the world until the 19th century. The legacy of Euclid outlived the scientist by as much as 200 centuries, and served as a source of inspiration for such personalities as, for example, Abraham Lincoln. Rumor has it that Lincoln always carried the Principia superstitiously with him, and in all his speeches he quoted the works of Euclid. Even after the death of a scientist, mathematician different countries continued to prove theorems and publish works under his name. In general, in those times when knowledge was closed to the masses, Euclid logically and scientifically created the format of ancient mathematics, which today is known to the world under the name of "Euclidean geometry".

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Kupchin's Youth Readings “Science. Creation. Search".
Section "Mathematics"

"Euclid and his contribution to science"

The work was done by a student of 6 "B" class
Surovegin Nikolai
Head: Vasilyeva
Daria Gennadievna

Saint Petersburg 2008

I. Introduction…………………………………….…3

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

III. Biography of Euclid……………………….….5

IV. Euclid's algorithm……………………………8

V. Axiomatics ....……………………………….11

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

VII. Started…………………………………………19

VIII. Tasks from the beginnings of Euclid…………………...22

IX. Problem solving………………………………..23

X. Links to information sources…...24

XI. Conclusion…………………………………..25

I Introduction

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

Chapter II.
Mathematics in Ancient Greece

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

Chapter III Biography of Euclid

EUCLID (Euclidc.356-300 BC)

BIOGRAPHY

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

ADVANCES IN MATHEMATICS

The main works of Euclid's "Beginnings" (Latinized name - "Elements") contains a presentation of planimetry, stereometry and a number of issues in number theory, algebra, general theory of relations and a method for determining areas and volumes, including elements of limits (Exhaustion method). In the Elements, Euclid summed up all the previous achievements of Greek mathematics and created the foundation for its further development. The historical significance of the "Beginnings" of Euclid lies in the fact that in them an attempt was made for the first time to construct a logical geometry on the basis of axiomatics. The main shortcoming of Euclid's axiomatics should be considered its incompleteness; there are no axioms of continuity, movement and order, so Euclid often had to appeal to intuition, to trust the eye. Books XIV and XV are later additions, but whether the first thirteen books are the work of one person or of a school led by Euclid is not known. Since 1482 "Beginnings" Euclid withstood more than 500 editions. in all languages ​​of the world.

"Beginnings"

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

Book I is preceded by definitions of concepts used in what follows. They are intuitive because they are defined in terms of physical reality: "A point is that which has no parts." "A line is length without breadth." "A straight line is one that is equally spaced in relation to the points on it." "A surface is that which has only length and width," etc.

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

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

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

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

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

Book XI is devoted to stereometry. In Book XII, which also probably goes back to Eudoxus, the areas of curvilinear figures are compared with the areas of polygons by means of the Method of exhaustion. The subject of Book XIII is the construction of regular polyhedra. The construction of the Platonic solids, which, apparently, ends the "Beginnings", gave reason to rank Euclid among the followers of Plato's philosophy.

AREAS OF INTEREST

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

MERITS OF EUCLID

Euclid's prime number theorem: The set of prime numbers is infinite (Euclid's Elements, Book IX, Theorem 20). More accurate quantitative information about the set of primes in the natural series is contained in the Chebyshev theorem on primes and asymptotics. the law of distribution of prime numbers.

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

Euclidean field - an ordered field in which every positive element is a square. For example, the field R of real numbers is F. p. The field Q of rational numbers is not F. p. L. Popov.

Euclidean space - a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, an E. n.

Chapter IV Euclid's algorithm

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

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

Euclid's algorithm for integers

Let a and b are integers not simultaneously equal to zero, and a sequence of numbers

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

a = bq 0 + r 1

b = r 1q 1 + r 2

r 1 = r 2q 2 + r 3

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

Correctness this algorithm follows from the following two statements:

Let a = bq + r, then ( a,b) = (b,r). (0,r) = r. for any non-zero r. Extended Euclid's Algorithm and Bezout's Relation

Formulas for ri can be rewritten like this:

r 1 = a + b(- q 0)

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

margin-top:0cm" type="disc"> Relation a / b admits a continued fraction representation:

.

Attitude - t / s, in the extended Euclidean algorithm can be represented as a continued fraction:

.

Variations and Generalizations

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

Accelerated versions of the algorithm

One of the methods for speeding up the integer Euclid algorithm is to choose symmetrical balance:

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

ChapterV.
axiomatics

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

Axiomatization theories - an explicit indication of a finite set of axioms. Statements following from axioms are called theorems.

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

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

For the first time the term "axiom" is found in Aristo-322 BC. BC) and passed into mathematics from the philosophers of ancient Greece. Euclid distinguishes between the concepts of "postulate" and "axiom" without explaining their differences. Since the time of Boethius, postulates have been translated as requirements (petitio), axioms as general concepts. Originally the word "axiom" had the meaning of "truth evident in itself". In different manuscripts of the Beginnings of Euclid, the division of statements into axioms and postulates is different, their order does not match. Probably the scribes held different views on the difference between these concepts.

ChapterVI. Euclidean geometry and the V postulate

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

Axiomatization

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

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

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

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

Euclid's axiom of parallelism or fifth postulate- one of the axioms underlying classical planimetry. First cited in Euclid's Elements.

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

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

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

If the sum of internal angles with a common side formed by two lines at the intersection of their third, on one of the sides of the secant is less than 180 °, then these lines intersect, and, moreover, on the same side of the secant.

In school textbooks, another formulation is usually given, equivalent (equivalent) to Postulate V and belonging to Proclus:

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

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

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

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

proof attempts

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

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

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

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

After the decline of ancient culture, postulate V was taken up by the mathematicians of the countries of Islam. The proof of al-Abbas al-Jawhari, a student of al-Khwarizmi (IX century), implicitly implied: if at the intersection of two lines by any third, the cross-lying angles are equal, then the same occurs when the same two lines intersect any other. And this assumption is equivalent to the fifth postulate.

Thabit ibn Qurra (9th century) gave 2 proofs; in the first he relies on the assumption that if two lines move away from each other on one side, they necessarily approach on the other side. In the second, it proceeds from the existence of equidistant straight lines, and Ibn Kurra is trying to derive this fact from the idea of ​​\u200b\u200b"simple movement", that is, about uniform movement at a fixed distance from a straight line (it seems obvious to him that the trajectory of such a movement is also a straight line). Each of the two mentioned statements of Ibn Qurra is equivalent to the fifth postulate.

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Saccheri's composition

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

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

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

Apparently, Saccheri felt the groundlessness of this "evidence", because the research is ongoing. He considers the equidistant - the locus of points of the plane, equidistant from the straight line; unlike his predecessors, Saccheri knows that in this case it is not a straight line at all. However, when calculating the length of its arc, Saccheri makes a mistake and comes to a real contradiction, after which he ends the study and declares with relief that he " uprooted this pernicious hypothesis».

In the second half of the 18th century, more than 50 works on the theory of parallels were published. In a review of those years (), more than 30 attempts to prove the V postulate are examined and their fallacy is proved. A well-known German mathematician and physicist, with whom Klugel corresponded, also became interested in the problem; his Theory of Parallel Lines was published posthumously in 1786.

Spherical geometry: all lines intersect

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

In his book, Lambert astutely noted:

It seems to me very remarkable that the second hypothesis [of an obtuse angle] is justified if instead of flat triangles we take spherical ones. I should almost have to draw a conclusion from this - the conclusion that the third hypothesis holds on some imaginary sphere. In any case, there must be a reason why it is far from being so easily refuted on the plane as it could be done with respect to the second hypothesis.

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

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

Neither Lobachevsky nor Bolyai were able to prove the consistency of the new geometry - at that time mathematics did not yet have the means necessary for this. Only 40 years later, the Klein model and other models that implement the axiomatics of Lobachevsky geometry on the basis of Euclidean geometry appeared. These models convincingly prove that the denial of the fifth postulate does not contradict the rest of the axioms of geometry; this implies that the V postulate is independent of the other axioms, and it is impossible to prove it. The centuries-old drama of ideas is over.

Chapter VII. The beginning of Euclid.

Greek text Beginnings.

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

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Euclid (otherwise 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 Euclid's teachers in Athens were students of Plato, and during the reign of Ptolemy I (306-283 BC) he taught at the Alexandria Academy. Euclid is the first mathematician of the Alexandrian school. Euclid is the author of a number of works on astronomy, optics, music, etc. Arab authors also attribute various treatises on mechanics to Euclid, including works on weights and on determining specific gravity. Euclid died between 275 and 270 BC. e.

Principles of Euclid

The main work of Euclid is called the Beginnings. Books with the same name, in which all the basic facts of geometry and theoretical arithmetic were consistently stated, were compiled earlier by Hippocrates of Chios, Leontes and Theeudius. However, the Elements of Euclid forced all these works out of use and for more than two millennia remained the basic textbook of geometry. In creating his textbook, Euclid included much of what had been 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 specify basic constructions (for example, "it is required that a line can be drawn through any two points"), and axioms give general rules for inference when operating with quantities (for example, "if two quantities are equal to a third, they are equal to between themselves").

Book I studies the properties of triangles and parallelograms; this book is crowned by the famous Pythagorean theorem for right triangles. Book II, dating back to the Pythagoreans, is devoted to the so-called "geometric algebra". Books III and IV deal with the geometry of circles, as well as inscribed and circumscribed polygons; when working on these books, Euclid could use the writings of Hippocrates of Chios. Book V introduces the general theory of proportions built by Eudoxus of Cnidus, and in Book VI it is applied to the theory of similar figures. Books VII-IX are devoted to the theory of numbers and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. These books deal with theorems on proportions and geometric progressions, introduce a method for finding the greatest common divisor of two numbers (now known as Euclid's algorithm), construct even perfect numbers, and prove the infinity of the set of primes. In the X book, which is the most voluminous and complex part of the Beginnings, a classification of irrationalities is built; it is possible that its author is Theaetetus of Athens. Book XI contains the fundamentals of stereometry. In Book XII, using the exhaustion method, theorems are proved on 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 developed by Theaetetus of Athens.

In the manuscripts that have come down to us, two more have been 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 (beginning of the 6th century AD).

The beginnings provide a common basis for subsequent geometric treatises by Archimedes, Apollonius and other ancient authors; the propositions proved in them are considered to be well known. Commentaries on the Principles in antiquity were composed by Heron, Porfiry, Pappus, Proclus, Simplicius. A commentary by Proclus to Book I has been preserved, as well as a commentary by Pappus to 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 the science of the New Age, the Beginnings also played an important ideological role. They remained an example of a mathematical treatise, strictly and systematically expounding the main provisions of a particular mathematical science.

The second work of Euclid after the "Beginnings" is usually called "Data" - an introduction to geometric analysis. Euclid also owns “Phenomena” devoted to elementary spherical astronomy, “Optics” and “Katoptrik”, 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 derived from precisely formulated physical hypotheses and mathematical postulates. Many of Euclid's works have been lost; we know of their existence in the past only through references in the writings of other authors.

Euclid, son of Naucrates, known by the name of "Geometer", a scientist of the old times, Greek by origin, Syrian by place of residence, originally from Tyre.

One of the legends tells 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 an 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 popular, has come down to us.

King Ptolemy I, in order to glorify his state, attracted scientists and poets to the country, creating for them the temple of the muses - Museion. There were study rooms, a botanical and zoological garden, an astronomical study, an astronomical tower, rooms for solitary 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 its students.

It was in Alexandria that Euclid founded a mathematical school and wrote a great work on geometry, united under the general title "Elements" - 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 all these were separate fragments, not a single logical scheme.

It is usually said of Euclid's Elements that, after the Bible, it is the most popular written monument of antiquity. The book has a very interesting history. For two thousand years, it was a reference book for schoolchildren, used as initial course geometry. The Elements were extremely popular, and many copies were made of them by industrious scribes in various cities and countries. Later, the "Beginnings" moved from papyrus to parchment, and then to paper. Over the course of four centuries, the "Principles" were published 2500 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 "Elements" of Euclid were thoroughly studied by the Arabs, and later by European scientists. They have been translated into the main 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) the division theorem with a remainder. Secondly, he came up with "Euclid's algorithm" - fast way finding 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 prime numbers - and proved that their set is infinite.

Euclid or Euclid(other Greek. Εὐκλείδης , from "good fame", heyday - about 300 BC. BC) - ancient Greek mathematician, author of the first theoretical treatise on mathematics that has come down to us. Biographical information about Euclid is extremely scarce. The only thing that can be considered reliable is that scientific activity flowed in Alexandria in the III century. BC e.

Biography

It is customary to attribute to the most reliable information about the life of Euclid the little that is given in the comments of Proclus to the first book. Began Euclid (although it should be taken into account that Proclus lived almost 800 years after Euclid). Noting that “mathematicians who wrote on the history” did not bring 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, “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 the portrait of Euclid can be gleaned from Pappus and Stobeus. Papp reports that Euclid was gentle and amiable with everyone who could contribute even in the slightest degree to the development of mathematical sciences, and Stobaeus relates another anecdote about Euclid. Having begun the study of geometry and having analyzed the first theorem, one young man asked Euclid: “And what will be the benefit to me 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 doubtful, as a similar story is told about Plato.

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

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

Euclid, son of Naucrates, known under the name of "Geometer", a scientist of the old time, Greek by origin, Syrian by residence, originally from Tyre ...

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

« Beginnings» Euclid

Euclid's main work is called Started. Books with the same title, which successively presented all the basic facts of geometry and theoretical arithmetic, were compiled earlier by Hippocrates of Chios, Leontes and Theeudius. However Beginnings Euclid pushed all these writings out of use and for more than two millennia remained the basic textbook of geometry. In creating his textbook, Euclid included much of what had been created by his predecessors, processing this material and bringing it together.

Beginnings consists 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 line can be drawn through any two points"), and axioms - general rules for inference when operating with quantities (for example, "if two quantities are equal to a third, they are equal between yourself").

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

Book I studies the properties of triangles and parallelograms; this book is crowned by the famous Pythagorean theorem for right triangles. Book II, dating back to the Pythagoreans, is devoted to the so-called "geometric algebra". Books III and IV deal with the geometry of circles, as well as inscribed and circumscribed polygons; when working on these books, Euclid could use the writings of Hippocrates of Chios. Book V introduces the general theory of proportions built by Eudoxus of Cnidus, and in Book VI it is applied to the theory of similar figures. Books VII-IX are devoted to the theory of numbers and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. These books deal with theorems on proportions and geometric progressions, introduce a method for finding the greatest common divisor of two numbers (now known as Euclid's algorithm), construct even perfect numbers, and prove the infinity of the set of primes. 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 fundamentals of stereometry. In Book XII, using the exhaustion method, theorems are proved on 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 Theaetetus of Athens.

In the manuscripts that have come down to us, two more have been 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, the builder of the church of St. Sophia in Constantinople (beginning of the 6th century AD).

Beginnings provide a common basis for subsequent geometric treatises by Archimedes, Apollonius, and other ancient authors; the propositions proved in them are considered to be well known. Comments on Beginnings in antiquity they were Heron, Porphyry, Pappus, Proclus, Simplicius. A commentary by Proclus to Book I has been preserved, as well as a commentary by Pappus to 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 an example of a mathematical treatise, strictly and systematically expounding the main provisions of a particular mathematical science.

Other works by Euclid

From other writings of Euclid survived:

• Data (δεδομένα ) - about what is needed to set the figure;
• About division (περὶ διαιρέσεων ) - preserved partially and only in 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 the rectilinear propagation of light.

The short descriptions are:

• porisms (πορίσματα ) - about the conditions that determine the curves;
• Conic 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 Nephtonian books selected and abbreviated in eight books through the professor of mathematics A. Farhvarson. / Per. from lat. I. Satarova. SPb., 1739. 284 pages.
• 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 pp.
• Euclidean Elements eight books, namely: 1st, 2nd, 3rd, 4th, 5th, 6th, 11th and 12th. / Per. from Greek SPb.,