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EUCLID (about 365 - 300 BC)
Gallery of great mathematicians
Prepared by the teacher of mathematics MOU secondary school No. 36 of Kaliningrad Kovalchuk Larisa Leonidovna
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Almost nothing is known about the life of this scientist. Only a few legends about him have come down to us. The first commentator on the "Beginnings" Proclus (5th century AD) could not indicate where and when Euclid was born and died. According to Proclus, “this learned man” lived in the era of the reign of Ptolemy I. Some biographical data are preserved on the pages of an Arabic manuscript of the XII century: Syrian, native of Tyre.
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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.
slide 4
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.
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It was in Alexandria that Euclid founded a mathematical school and wrote a great work on geometry, united under the general title "Beginnings" - the main work of his life. It is believed to have been written around 325 BC. The predecessors of Euclid - Thales, Pythagoras, Aristotle and others did a lot for the development of geometry. But all these were separate fragments, not a single logical scheme.
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Both contemporaries and followers of Euclid were attracted by the systematic and logical nature of the information presented. "Beginnings" consists of thirteen books, built according to a single logical scheme. Each of the thirteen books begins with a definition of the concepts (point, line, plane, figure, etc.) that are used in it, and then, based on a small number of basic provisions (5 axioms and 5 postulates), accepted without proof, the whole system is built geometry.
Slide 7
At that time, the development of science did not imply the existence of methods of practical mathematics. Books I-IV covered geometry, and their content was traced back to the works of the Pythagorean school. In book V, the doctrine of proportions was developed, which was adjacent to Eudoxus of Cnidus. Books VII-IX contained the doctrine of numbers, representing the development of the Pythagorean primary sources. Books X-XII contain definitions of areas in the plane and space (stereometry), the theory of irrationality (especially in Book X); book XIII contains studies of regular bodies, going back to Theaetetus.
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Raphael Santi, Euclid, detail 1508-11, fresco "School of Athens" Stanz della Senyatura, Vatican, Rome, Italy
Slide 9
Euclid's "Elements" is a presentation of that geometry, which is known to this day under the name of Euclidean geometry. It describes the metric properties of the space that modern science calls the Euclidean space. Euclidean space is the arena of physical phenomena of classical physics, the foundations of which were laid by Galileo and Newton. This space is empty, boundless, isotropic, having three dimensions. Euclid gave mathematical certainty to the atomistic idea of empty space in which atoms move. Euclid's simplest geometric object is the point, which he defines as something that has no parts. In other words, a point is an indivisible atom of space.
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The infinity of space is characterized by three postulates: "A straight line can be drawn from any point to any point." "A bounded straight line can be continuously extended along a straight line." "From every center and every solution a circle can be described."
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The doctrine of parallels and the famous fifth postulate (“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”) define the properties of Euclidean space and its geometry, different from non-Euclidean geometries.
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It is usually said about the "Principles" 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 an elementary course in geometry. The Elements were extremely popular, and many copies were made of them by industrious scribes in various cities and countries. Later, the "Beginnings" were transferred from papyrus to parchment, and then to paper. Over the course of four centuries, the "Beginnings" were published 2,500 times: on average, 6-7 editions were published annually. Until the 20th century, the book was considered the main textbook on geometry, not only for schools, but also for universities.
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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 translation into English, dating back to 1570, was made by Henry Billingway, the London merchant Euclid owns partly preserved, partly reconstructed later mathematical works It was he who introduced the algorithm for obtaining the greatest common divisor two arbitrarily taken natural numbers and an algorithm called the "Eratosthenes account" for finding prime numbers from a given number.
Slide 14
Euclid laid the foundations of geometric optics, which he outlined in the works "Optics" and "Katoptrik". The basic concept of geometric optics is a rectilinear light beam. Euclid argued that the light beam comes from the eye (the theory of visual rays), which is not essential for geometric constructions. He knows the law of reflection and the focusing action of a concave spherical mirror, although he still cannot determine the exact position of the focus. In any case, in the history of physics, the name of Euclid as the founder of geometric optics has taken its proper place.
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In Euclid, we also find a description of the monochord - a single-string instrument for determining the pitch of a string and its parts. It is believed that Pythagoras invented the monochord, and Euclid only described it (“Division of the Canon”, III century BC). Euclid, with his characteristic passion, took up the numeral system of interval relations. The invention of the monochord was significant for the development of music. Gradually, instead of one string, two or three began to be used. This was the beginning of the creation of keyboard instruments, first the harpsichord, then the piano, and mathematics became the root cause of the appearance of these musical instruments.
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Of course, all the features of the Euclidean space were not discovered immediately, but as a result of the centuries-old work of scientific thought, but the starting point of this work was the "Beginnings" of Euclid. Knowledge of the foundations of Euclidean geometry is now a necessary element of general education throughout the world.
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http://biographera.net/biography.php?id=50 http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euclid.html
The outstanding ancient Greek mathematician Euclid was born in Megara, a small Greek town. We know very little about his life, even the date of birth and death of this person is unknown. Usually only the fourth century BC is indicated, when he was born, and the third century BC, the heyday of his activities in Alexandria, the capital of Egypt under the Greco-Macedonian Ptolemaic dynasty. In the ancient world, the Ptolemies were unparalleled in their patronage of scientists, writers, inventors and poets. It is known that he was a student of Plato.
One day, King Ptolemy asked Euclid if there was another, less difficult way of knowing geometry than the one that the scientist outlined in his "Principles". Euclid replied: O king, there are no royal roads in geometry ».
- For a long time, scientists believed that there was no specific historical person, that a group of mathematicians was hiding under the name of Euclid. However, evidence of its existence was found in a found manuscript of the 12th century. Euclid ended up in Alexandria as a teacher of Museion, i.e. literally "the abode of the Muses", but in fact - the prototype of future European universities. In this magnificent city, Euclid created his work "Beginnings" (or "Elements" in a Latinized form). The fifteen books of the "Principles" contain almost all the most important achievements of ancient mathematics. For more than two thousand years, the Euclidean essay remained the main work on elementary mathematics. But the achievement of Euclid is not only in the fact that he discovered laws and theorems, but also in the fact that the great mathematician brought scattered and extensive theoretical material into a system and arranged it in such a sequence that each theorem followed from the previous one. He gave the first system of axioms - statements accepted without proof. The fact that mathematics is called the most exact of the sciences is a considerable merit of Euclid.
- And now let's talk about what exactly Euclid's discoveries were.
- The basics of geometric algebra (the science of calculating segments and areas) were outlined in I book"Started". There, segments are considered and arithmetic operations on them are defined. For example, two segments were added, attaching one to the other, subtracted, removing from the larger segment a part equal to the smaller one. The calculus defined in geometric algebra was "graded". The first stage consisted of segments, the second - areas, the third - volumes. The tools with which it was allowed to make constructions in geometric algebra were compasses and a ruler.
- AT Book II the main properties of triangles, rectangles, parallelograms are considered and their areas are compared. The book ends with the Pythagorean theorem.
- AT Book III the properties of the circle, its tangents and chords are considered (these problems were studied by Hippocrates of Chios in the second half of the 5th century BC).
In 1739, the book "Beginnings" was translated into Russian. Here is the first page of the first book.
- AT Book IV are regular polygons. AT Book V the general theory of the relations of magnitudes, created by Eudoxus of Cnidus, is given; it can be regarded as a prototype of the theory of real numbers, developed only in the second half of the 19th century. The general theory of relations is the basis of the doctrine of similarity (book VI) and the method of exhaustion (book VII), also dating back to Eudoxus. AT books VII-IX the beginnings of number theory based on the algorithm for finding the greatest common divisor or Euclid's algorithm are outlined. These books include the theory of divisibility, including theorems on the uniqueness of factoring an integer into prime factors and on the infinity of the number of primes; here the doctrine of the ratio of integers is also expounded, similar to the theory of rational (positive) numbers. AT book X a classification of quadratic and biquadratic irrationalities is given and some rules for their transformation are substantiated. The results of Book X are applied in Book XIII to find the edge lengths of regular polyhedra. significant portion books X and XIII(probably also VII) belongs to Theaetetus (beginning of the 4th century BC). AT Book XI outlines the basics of stereometry.
- AT Book XII the ratio of the areas of two circles and the ratio of the volumes of a pyramid and a prism, a cone and a cylinder are determined using the exhaustion method. These theorems were first proved by Eudoxus.
- Finally, in Book XIII the ratio of the volumes of two balls is determined, five regular polyhedra are constructed, and it is proved that there are no other regular bodies.
- Subsequent Greek mathematicians added to Euclid's Elements Books XIV and XV, which did not belong to Euclid. They are often and now published together with the main text of the "Beginnings". There, segments are considered and arithmetic operations on them are defined.
Fragment of the oldest papyrus with diagrams from Euclid's "Elements of Geometry"
- Citadel (medieval fortress) built in XII century
Al-Mursi Abul Abbas's Mosque Alexandria .
Hurghada. Palace 1000 and 1 night. Alexandria
alexandria bay
Alimov N. G. Value and relation in Euclid. Historical and Mathematical Research, vol. 8, 1955, p. 573-619. Bashmakova I. G. Arithmetic books of the "Beginnings" of Euclid. Historical and Mathematical Research, vol. 1, 1948, p. 296-328. Van der Waerden B.L. Awakening Science. M .: Fizmatgiz, 1959. Vygodsky M. Ya. "Beginnings" of Euclid. Historical and Mathematical Research, vol. 1, 1948, p. 217-295. Glebkin V.V. Science in the context of culture: ("Beginnings" of 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: Kagan VF Foundations of Geometry. Part 1. M., 1949, p. 28-110. Raik A.E. The tenth book of Euclid's "Beginnings". 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. Zeiten GG History of mathematics in antiquity and in 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, vol. 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. ?????, no. 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 arithmetiqu's 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. Euklid. Leipzig: Teubner, 1987.
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The life and work of Euclid Euclid (presumably 330-277 BC) is a mathematician of the Alexandrian school of ancient Greece, the author of the first treatise on mathematics that has come down to us.
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Five postulates of Euclid From any point to any other point it is possible to draw only one straight line. A limited straight line can be continued continuously in a straight line. From any center and any solution it is possible to describe a circle. All right angles are equal to each other. If a line falling on two lines forms interior and on one side angles less than two right angles, then these two lines extended indefinitely meet on the side where the angles are less than two
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Fifth postulate If a line falling on two lines forms interior and on one side angles less than two lines, then these two lines extended indefinitely meet on the side where the angles are less than two lines.
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The fifth postulate of parallels was formulated by: Proclus (411 - 485 BC) Euclid (325 - 265 BC) Archimedes (287 - 212 BC) Ptolemy (85 - 165 BC) Wallis (1663) Legendre (1794, 1823), and even the famous poet Omar Khayyam But the "godfather" of non-Euclidean geometry turned out to be an Italian monk who taught mathematics and grammar Girolamo Saccheri, known for his dying treatise (1766): "Euclid, cleansed of all stains" .
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9 axioms of Euclid Equals to the same thing are equal and to each other If equals are added to equals, then the integers will be equal If equals are subtracted from equals, then the remainders will be equal If equals are added to unequals, then the integers will not be equal
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9 axioms of Euclid (continued) Doubled of the same are equal to each other Halves of the same are equal to each other Combining one with the other are equal to each other The whole is greater than the part Two straight lines do not contain space
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Conclusion Euclid made three significant discoveries in arithmetic. First, he formulated (without proof) the division theorem with a remainder. Secondly, he came up with "Euclid's algorithm" - a quick way to find the greatest common divisor of numbers or a 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. But is it true that any integer can be decomposed into a product of prime numbers in a unique way? Euclid failed to prove this - although he had all the means necessary for this.
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slide 1
EUCLID (c. 365 - 300 BC) Gallery of great mathematicians Prepared by the mathematics teacher of the Moscow School of Education No. 36 of Kaliningrad Kovalchuk Larisa Leonidovnaslide 2
Almost nothing is known about the life of this scientist. Only a few legends about him have come down to us. The first commentator on the "Beginnings" Proclus (5th century AD) could not indicate where and when Euclid was born and died. According to Proclus, “this learned man” lived in the era of the reign of Ptolemy I. Some biographical data are preserved on the pages of an Arabic manuscript of the XII century: Syrian, native of Tyre.
slide 3
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.
slide 4
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.
slide 5
It was in Alexandria that Euclid founded a mathematical school and wrote a great work on geometry, united under the general title "Beginnings" - the main work of his life. It is believed to have been written around 325 BC. The predecessors of Euclid - Thales, Pythagoras, Aristotle and others did a lot for the development of geometry. But all these were separate fragments, not a single logical scheme.
slide 6
Both contemporaries and followers of Euclid were attracted by the systematic and logical nature of the information presented. "Beginnings" consists of thirteen books, built according to a single logical scheme. Each of the thirteen books begins with a definition of the concepts (point, line, plane, figure, etc.) that are used in it, and then, based on a small number of basic provisions (5 axioms and 5 postulates), accepted without proof, the whole system is built geometry.
Slide 7
At that time, the development of science did not imply the existence of methods of practical mathematics. Books I-IV covered geometry, and their content was traced back to the works of the Pythagorean school. In book V, the doctrine of proportions was developed, which was adjacent to Eudoxus of Cnidus. Books VII-IX contained the doctrine of numbers, representing the development of the Pythagorean primary sources. Books X-XII contain definitions of areas in the plane and space (stereometry), the theory of irrationality (especially in Book X); book XIII contains studies of regular bodies, going back to Theaetetus.
Slide 8
Raphael Santi, Euclid, detail 1508-11, fresco "School of Athens" Stanz della Senyatura, Vatican, Rome, Italy
Slide 9
Euclid's "Elements" is a presentation of that geometry, which is known to this day under the name of Euclidean geometry. It describes the metric properties of the space that modern science calls the Euclidean space. Euclidean space is the arena of physical phenomena of classical physics, the foundations of which were laid by Galileo and Newton. This space is empty, boundless, isotropic, having three dimensions. Euclid gave mathematical certainty to the atomistic idea of empty space in which atoms move. Euclid's simplest geometric object is the point, which he defines as something that has no parts. In other words, a point is an indivisible atom of space.
slide 10
The infinity of space is characterized by three postulates: "A straight line can be drawn from any point to any point." "A bounded straight line can be continuously extended along a straight line." "From every center and every solution a circle can be described."
slide 11
The doctrine of parallels and the famous fifth postulate (“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”) define the properties of Euclidean space and its geometry, different from non-Euclidean geometries.
slide 12
It is usually said about the "Principles" 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 an elementary course in geometry. The Elements were extremely popular, and many copies were made of them by industrious scribes in various cities and countries. Later, the "Beginnings" were transferred from papyrus to parchment, and then to paper. Over the course of four centuries, the "Beginnings" were published 2,500 times: on average, 6-7 editions were published annually. Until the 20th century, the book was considered the main textbook on geometry, not only for schools, but also for universities.
slide 13
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 translation into English, dating back to 1570, was made by Henry Billingway, the London merchant Euclid owns partly preserved, partly reconstructed later mathematical works It was he who introduced the algorithm for obtaining the greatest common divisor two arbitrarily taken natural numbers and an algorithm called the "Eratosthenes account" for finding prime numbers from a given number.
slide 14
Euclid laid the foundations of geometric optics, which he outlined in the works "Optics" and "Katoptrik". The basic concept of geometric optics is a rectilinear light beam. Euclid argued that the light beam comes from the eye (the theory of visual rays), which is not essential for geometric constructions. He knows the law of reflection and the focusing action of a concave spherical mirror, although he still cannot determine the exact position of the focus. In any case, in the history of physics, the name of Euclid as the founder of geometric optics has taken its proper place.
slide 15
In Euclid, we also find a description of the monochord - a single-string instrument for determining the pitch of a string and its parts. It is believed that Pythagoras invented the monochord, and Euclid only described it (“Division of the Canon”, III century BC). Euclid, with his characteristic passion, took up the numeral system of interval relations. The invention of the monochord was significant for the development of music. Gradually, instead of one string, two or three began to be used. This was the beginning of the creation of keyboard instruments, first the harpsichord, then the piano, and mathematics became the root cause of the appearance of these musical instruments. http://biographera.net/biography.php?id=50 http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euclid.html Information sources:
