Euclid
The project was carried out
7B grade student
Filippova Anna
Euclid- ancient Greek mathematician, author of the first theoretical treatise on mathematics that has reached us. Biographical information about Euclid is extremely scarce. The only thing that can be considered reliable is that his scientific activity took place in Alexandria in the 3rd century. BC e.
Euclid's Elements
Euclid's main work is called
Beginnings. Books with the same title
which consistently set out
all the basic facts of geometry and
theoretical arithmetic, compiled
previously Hippocrates of Chios , Leontes And
Fevdiem. However Beginnings Euclid
displaced all these writings from
everyday life and for more than two
remained basic for millennia
geometry textbook. Creating your
textbook, Euclid included a lot in it
from what was created by him
predecessors, having processed this
material and bringing it together
Beginnings consist of thirteen books. The first and some other books are preceded by a list of definitions. The first book is also preceded by a list of postulates and axioms. Usually, postulates define basic constructions (for example, “it is required that a straight line can be drawn through any two points”), and axioms- general rules of inference when operating with quantities (for example, “if two quantities are equal to a third, they are equal to each other”).
In Book I, the properties of triangles and parallelograms are studied; This book is crowned with the famous Pythagorean theorem for right triangles. Book II, going back to the Pythagoreans, is devoted to the so-called “geometric algebra”. Books III and IV describe the geometry of circles, as well as inscribed and circumscribed polygons; when working on these books, Euclid could have used the works Hippocrates of Chios
Book V introduces the general theory of proportions, built Eudoxus of Cnidus, and in Book VI it is attached to the theory of similar figures. Books VII-IX are devoted to number theory and go back to the Pythagoreans; the author of Book VIII may have been Archytas of Tarentum. These books examine theorems on proportions and geometric progressions, introduce a method for finding the greatest common divisor of two numbers, and construct even perfect numbers, the infinity of the set is proved prime numbers. In the X book, which is the most voluminous and complex part Began, a classification of irrationalities is constructed; it is possible that its author is Theaetetus of Athens .
Book XI contains the basics of stereometry. In the XII book, using the method of exhaustion, theorems on the ratios of the areas of circles, as well as the volumes of pyramids and cones are proved; The author of this book is admittedly Eudoxus of Cnidus. Finally, Book XIII is devoted to the construction of five regular polyhedra; it is believed that some of the constructions were developed Theaetetus of Athens.
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 man is unknown. Usually they indicate only the fourth century BC, 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 had no equal 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 understanding geometry than the one that the scientist outlined in his “Elements.” Euclid replied: " O king, in geometry there are no royal roads ».
- For a long time, scientists believed that there was no specific historical figure, that a group of mathematicians was hiding under the name of Euclid. However, evidence of its existence was found in a 12th-century manuscript that was found. Euclid ended up in Alexandria as a teacher at Museion, i.e. literally “the abode of the Muses”, and in fact - the prototype of future European universities. In this magnificent city, Euclid created his work “Elements” (or “Elements” in Latinized form). The fifteen books of the Elements contain almost all the most important achievements of ancient mathematics. For more than two thousand years, Euclid's work remained the main work on elementary mathematics. But Euclid’s achievement lies not only in the fact that he discovered laws and theorems, but also in the fact that the great mathematician brought into a system disparate and extensive theoretical material 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 sciences is a considerable merit of Euclid.
- Now let’s talk about what exactly Euclid’s discoveries were.
- The basics of geometric algebra (the science of calculating segments and areas) were presented in Book I"Began". There segments are considered and arithmetic operations on them are defined. For example, two segments were added by placing one next to the other, and subtracted by removing from the larger segment a part equal to the smaller one. Calculus, defined in geometric algebra, was "echelon". The first stage consisted of segments, the second - areas, the third - volumes. The tools with which it was allowed to carry out constructions in geometric algebra were the compass and ruler.
- IN book II The basic properties of triangles, rectangles, parallelograms are considered and their areas are compared. The book ends with the Pythagorean theorem.
- IN book III the properties of the circle, its tangents and chords are considered (these problems were studied by Hippocrates of Chios in the 2nd half of the 5th century BC).
In 1739, the book “Beginnings” was translated into Russian. Here is the first page of the first book.
- IN book IV- regular polygons. IN book V the general theory of quantity relations created by Eudoxus of Cnidus is given; it can be considered as a prototype of the theory of real numbers, developed only in the 2nd 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. IN books VII-IX the beginnings of number theory are presented, based on the algorithm for finding the greatest common divisor or the Euclidean algorithm. These books include the theory of divisibility, including theorems on the uniqueness of the factorization of an integer into prime factors and on the infinity of the number of prime numbers; It also expounds a doctrine of the ratio of integers similar to the theory of rational (positive) numbers. IN 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 used in Book XIII to find the edge lengths of regular polyhedra. Substantial part books X and XIII(probably VII) belongs to Theaetetus (early 4th century BC). IN book XI the basics of stereometry are outlined.
- IN book XII Using the exhaustion method, 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. These theorems were first proven 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 even now published together with the main text of “Principles.” There segments are considered and arithmetic operations on them are defined.
Fragment of the oldest papyrus with diagrams from Euclid's Elements of Geometry
- The citadel (medieval fortress) was built in XII century
Al-Mursi Abul Abbas Mosque in Alexandria .
Hurghada. Palace 1000 and 1 night. Alexandria
Alexandria Bay
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EUCLID (c. 365 - 300 BC)
Gallery of Great Mathematicians
Prepared by mathematics teacher of Municipal Educational Institution 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 reached us. The first commentator on the Elements, Proclus (5th century AD), could not indicate where and when Euclid was born and died. According to Proclus, “this learned man” lived during the reign of Ptolemy I. Some biographical data was preserved on the pages of an Arabic manuscript of the 12th century: “Euclid, son of Naukrates, known under the name “Geometra”, a scientist of old times, Greek by origin, by residence Syrian, originally from Tyre.”
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One of the legends says that King Ptolemy decided to study geometry. But it turned out that this is not so easy to do. Then he called Euclid and asked him to show him an easy path to mathematics. “There is no royal road to geometry,” the scientist answered him. This is how this popular expression came to us in the form of a legend.
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King Ptolemy I, in order to exalt his state, attracted scientists and poets to the country, creating for them a temple of muses - Museion. There were study rooms, botanical and zoological gardens, an astronomical office, an astronomical tower, rooms for 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 “Elements” - the main work of his life. It is believed to have been written around 325 BC. Euclid's predecessors - Thales, Pythagoras, Aristotle and others - did a lot for the development of geometry. But all these were separate fragments, and 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. “Principles” 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 entire system is built geometry.
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At that time, the development of science did not imply the presence of methods of practical mathematics. Books I-IV covered geometry, their content going 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 Pythagorean primary sources. Books X-XII contain definitions of areas in 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 Segnatura, Vatican, Rome, Italy
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Euclid's "Principles" are an exposition of the geometry that is still known today under the name Euclidean geometry. It describes the metric properties of space, which modern science calls 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, limitless, 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 a 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.” “A circle can be described from any center and by any solution.”
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The doctrine of parallels and the famous fifth postulate (“If a straight line falling on two straight lines forms interior angles and on one side less than two right angles, then extended indefinitely these two straight lines will meet on the side where the angles are less than two right angles”) determine the properties of Euclidean space and its geometry, different from non-Euclidean geometries.
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It is usually said of the Elements that, after the Bible, it is the most popular written monument of antiquity. The book has its own, very remarkable history. For two thousand years it was a reference book for schoolchildren and was used as an initial course in geometry. The Elements were extremely popular, and many copies were made from them by industrious scribes in different cities and countries. Later, the “Principles” were transferred from papyrus to parchment, and then to paper. Over the course of four centuries, the “Principles” 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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Euclid's "Principles" were thoroughly studied by the Arabs and later by European scientists. They have been translated into major world languages. The first originals were printed in 1533 in Basel. It is curious that the first translation into English, dating back to 1570, was made by Henry Billingway, the London merchant Euclid owns partially preserved, partially reconstructed mathematical works. It was he who introduced the algorithm for obtaining the greatest common divisor two arbitrarily chosen natural numbers and an algorithm called “Eratosthenes’ counting” for finding prime numbers from a given number.
Slide 14
Euclid laid the foundations of geometric optics, which he outlined in his works “Optics” and “Catoptrics”. The basic concept of geometric optics is a rectilinear light beam. Euclid argued that a light ray comes from the eye (the theory of visual rays), which is not significant for geometric constructions. He knows the law of reflection and the focusing effect 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 a monochord - a single-string device for determining the pitch of a string and its parts. It is believed that the monochord was invented by Pythagoras, and Euclid only described it (“Division of the Canon,” 3rd century BC). Euclid, with his characteristic passion, took up the numerical system of interval relations. The invention of the monochord was important 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 the root cause of the appearance of these musical instruments was mathematics.
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Of course, all the features of Euclidean space were not discovered immediately, but as a result of centuries of work of scientific thought, but the starting point of this work was Euclid’s “Elements”. Knowledge of the basics 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
Euclid or Euclid is an ancient Greek mathematician. He gained worldwide fame thanks to his essay on the basics of mathematics, “Principia.” Biographical information about Euclid is extremely scarce. Almost nothing is known about the life of Euclid. Some biographical data has been preserved on the pages of an Arabic manuscript of the 12th century: “Euclid, son of Naukrates, known as Geometra, a scientist of old times, Greek by origin, Syrian by residence, originally from Tyre.” He was born in Athens and studied at the Academy. At the beginning of the 3rd century BC. moved to Alexandria and there founded a mathematical school and wrote his fundamental work for its students, united under the general title “Principles.” It was written around 325 BC. Euclid
In arithmetic, Euclid made three significant discoveries. First, he formulated (without proof) the theorem on division with remainder. Secondly, he came up with the “Euclidean algorithm” - a quick way to find the greatest common divisor of numbers or the common measure of segments (if they are commensurable). Finally, Euclid was the first to study the properties of prime numbers - and proved that their set is infinite.
Vatican Manuscript, vol. 1, 38v 39r. Euclid I prop. 47 (Pythagorean theorem). Of the works of Euclid that have come down to us, the most famous are the Elements, consisting of 15 books. The first four books of the Elements 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 later. They are intuitive in nature, since they are defined in terms of physical reality: “A point is something that has no parts.” “A line is length without width.” "A straight line is one that is equally located in relation to the points on it." “The surface is that which has only length and width,” etc.
Book II lays the foundations of the so-called geometric algebra, which dates 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, and Book IV studies regular polygons inscribed in a circle, as well as circumscribed around it. The theory of proportions, developed in Book V, applied equally well to commensurate quantities and 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 resorting to arithmetic, he did not assign numerical values to quantities.
In Book VI, the theory of proportions of Book V is applied to rectilinear figures, to geometry on the plane and, in particular, to similar figures, and “similar rectilinear figures are those which have angles equal in order, and sides at equal angles proportional.” Books VII, VIII and IX constitute a treatise on the theory of numbers; the theory of proportions is applied to numbers in them. Book VII defines the equality of ratios of integers, or, from a modern point of view, builds the theory of rational numbers. Of the many properties of numbers studied by Euclid (parity, divisibility, etc.), we cite, for example, proposition 20 of Book IX, which establishes the existence of an infinite set of “firsts,” i.e. prime numbers: “There are more prime numbers than any number of prime numbers offered.” His proof by contradiction can still be found in algebra textbooks.
Book X is difficult to read; it contains a classification of quadratic irrational quantities, which are represented there by geometric lines and rectangles. Here is how sentence 1 is formulated in Book X of Euclid’s Elements: “If two unequal quantities are given and from the larger a part greater than half is subtracted, and from the remainder again a part greater than half, and this is repeated constantly, then someday there remains a quantity that is less than the smaller of the given quantities." In modern language: If a and b are positive real numbers and a > b, then there always exists a natural number m such that mb > a. Euclid proved the validity of geometric transformations. b, then there always exists a 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 dates back to Eudoxus, the areas of curvilinear figures are compared with the areas of polygons using the Method of Exhaustion. The subject of Book XIII is the construction of regular polyhedra. The construction of the Platonic solids, with which, apparently, the “Principles” are completed, gave reason to classify Euclid as a follower of Plato’s philosophy.
The second work of Euclid after the Elements is usually called Data, an introduction to geometric analysis. Euclid also owns “Phenomena”, dedicated to elementary spherical astronomy, “Optics” and “Catoptrics”, 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” ( came to us in Arabic translation). The presentation in all these works, as in the Principia, 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 only know about their existence in the past through references in the works of other authors.
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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 with any solution it is possible to describe a circle. All right angles are equal to each other. If a straight line falling on two straight lines forms interior angles on one side that are less than two right angles, then the extensions of these two straight lines meet without limit on the side where the angles are less than two
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Fifth Postulate If a straight line falling on two straight lines forms interior angles on one side that are less than two right angles, then the extensions of these two straight lines meet without limit on the side where the angles are less than two right angles.
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The V parallel postulate 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, famous for his dying treatise (1766): “Euclidean, cleared of all stains” .
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9 axioms of Euclid Equals to the same thing are equal to each other If equals are added to equals, then the integers will also 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) Doubles of the same thing are equal to each other Halves of the same thing are equal to each other Complementaries of one and 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 In arithmetic, Euclid made three significant discoveries. First, he formulated (without proof) the theorem on division with remainder. Secondly, he came up with the “Euclidean algorithm” - a quick way to find the greatest common divisor of numbers or the common measure of segments (if they are commensurable). Finally, Euclid was the first to study the properties of 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 was unable to prove this, although he had all the necessary means for this.
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