Presentation on the topic Euclid. Presentation on the topic "Euclid and his "beginnings"

1 slide

2 slide

The first mentions of polyhedra are known three thousand years BC in Egypt and Babylon. But the theory of polyhedra is also a modern branch of mathematics. It is closely related to topology, graph theory, and is of great importance both for theoretical research in geometry and for practical applications in other branches of mathematics, for example, algebra, number theory, applied mathematics - linear programming, optimal control theory. They have a rich history, which is associated with the names of such scientists as Pythagoras, Euclid, Archimedes. polyhedra are distinguished by unusual properties, the most striking of which is formulated in Euler’s theorem on the number of faces, vertices and edges of a convex polyhedron: for any convex polyhedron the relation Г+В-Р=2 is true, where Г is the number of faces, В is the number of vertices, Р- the number of edges of a given polyhedron.

3 slide

4 slide

EUCLID, or EUCLID, is an ancient Greek mathematician, the author of the first theoretical treatises on mathematics that have reached us. Biographical information about the life and work of Euclid is extremely scarce. It is known that he was from Athens and was a student of Plato. Euclid's scientific activity took place in Alexandria (3rd century BC), and its heyday occurred during the reign of Ptolemy I Soter in Egypt. It is also known that Euclid was younger than Plato’s students (427-347 BC), but older than Archimedes (c. 287-212 BC), since, on the one hand, he was a Platonist and knew Plato’s philosophy well (that is why he ended the “Principles” with a presentation of the so-called Platonic solids, i.e., the five regular polyhedra), and on the other hand, his name is mentioned in the first of Archimedes’ two letters to Dositheus, “On the Ball and the Cylinder.”

5 slide

Geometric knowledge approximately equivalent to a modern high school course was presented 2200 years ago in Euclid’s Elements. Of course, the science of geometry outlined in the Elements could not have been created by one scientist. It is known that Euclid in his work relied on the works of dozens of predecessors, among whom were Thales and Pythagoras, Democritus and Hippocrates, Archytas, Theaetetus, Eudoxus and others. At the cost of great effort, based on individual geometric information accumulated over thousands of years in the practical activities of people, these Great scientists were able to bring geometric science to a high level of perfection over the course of 3-4 centuries. The historical merit of Euclid lies in the fact that, when creating his “Elements,” he combined the results of his predecessors, ordered and brought into one system the basic geometric knowledge of that time. For two thousand years, geometry was studied in the volume, order and style as it was presented in Euclid's Elements. Many elementary geometry textbooks around the world were (and many still are) merely a reworking of Euclid’s book. “Principia” has been a reference book for the greatest scientists for centuries.

6 slide

Euclid defines a pyramid as a solid figure bounded by planes that converge from one plane to one point.

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 “Principles.” 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 is not only that he discovered laws and theorems, but also 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. Before you is the first page of the book.

• IN book IV- regular polygons. IN book V the general theory of relations of quantities 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

Alimov N. G. Magnitude and relation in Euclid. Historical and mathematical studies, vol. 8, 1955, p. 573-619. Bashmakova I. G. Arithmetic books of Euclid’s Elements. Historical and mathematical studies, vol. 1, 1948, p. 296-328. Van der Waerden B. L. Awakening science. M.: Fizmatgiz, 1959. Vygodsky M. Ya. “Principles” of Euclid. Historical and mathematical studies, vol. 1, 1948, p. 217-295. Glebkin V.V. Science in the context of culture: (“Principles” of Euclid and “Jiu Zhang Xuan Shu”). M.: Interprax, 1994. 188 pp. 3000 copies. ISBN 5-85235-097-4 Kagan V.F. Euclid, his successors and commentators. In the book: Kagan V.F. Foundations of Geometry. Part 1. M., 1949, p. 28-110. Raik A. E. The tenth book of Euclid’s Elements. Historical and mathematical studies, vol. 1, 1948, p. 343-384. Rodin A.V. Mathematics of Euclid in the light of the philosophy of Plato and Aristotle. M.: Nauka, 2003. Tseyten G. G. History of mathematics in ancient times and in the Middle Ages. M.-L.: ONTI, 1938. Shchetnikov A.I. The second book of Euclid’s “Principles”: its mathematical content and structure. Historical and mathematical studies, vol. 12(47), 2007, p. 166-187. Shchetnikov A.I. The works of Plato and Aristotle as evidence of the formation of a system of mathematical definitions and axioms. ?????, vol. 1, 2007, p. 172-194. Artmann B. Euclid’s “Elements” and its prehistory. Apeiron, v. 24, 1991, p. 1-47. Brooker M.I.H., Connors J.R., Slee A.V. Euclid. CD-ROM. Melbourne, CSIRO-Publ., 1997. Burton H.E. The optics of Euclid. J. Opt. Soc. Amer., v. 35, 1945, p. 357-372. Itard J. Lex livres 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.

Slide 1

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

Slide 2

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."

Slide 3

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.

Slide 4

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.

Slide 5

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.

Slide 6

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.

Slide 7

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.

Slide 8

Raphael Santi, Euclid, detail 1508-11, fresco "School of Athens" Stanz della Segnatura, Vatican, Rome, Italy

Slide 9

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.

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.” “A circle can be described from any center and by any solution.”

Slide 11

The doctrine of parallels and the famous fifth postulate (“If a straight line falling on two straight lines forms interior angles 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.

Slide 12

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.

Slide 13

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.

Slide 15

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. http://biographera.net/biography.php?id=50 http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euclid.html Information sources:

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.