<Student’s name>

<Professor’s name>

Introduction

Euclid (near 365 - 300 BC) is a Greek mathematician, lived in Alexandria, in the 3rd BC. The main work "Elements" (15 books), containing the basics of ancient mathematics, elementary geometry, number theory, the general theory of relations and the method of determining areas and volumes, which included elements of the theory of limits, has had an enormous influence on the development of mathematics. He worked on astronomy, optics, music theory.

Information about the time and place of his birth did not reach us, but we know that Euclid lived in Alexandria and flourishing his activities were in Egypt during the reign of Ptolemy I Soter "The name Euclid associated formation of the Alexandrian mathematicians (geometric algebra) as a science.

Proclus in the comments to the first book "Principia" leads a famous anecdote about the issue, who allegedly asked Euclid, Ptolemy, " Is not there a more concise way of geometry than (the one that is presented) in the " Elements "? On Euclid allegedly replied: "there is no geometry royal road" (also describes a similar anecdote about Alexander and pupil of Eudoxus Menaechmus, so that it belongs to, apparently, one of the "stray scenes").

Euclid is the first mathematician of the Alexandrian school. His main work "Elements» (Στοιχεῖα) contains a summary of plane geometry, solid geometry and number theory, a number of issues; in which he summarized the prior development of the ancient Greek mathematician and created the foundation for the further development of mathematics.

## Elements

Extant writings of Euclid most famous "Elements", consisting of 15 books. In the 1st book formulated assumptions geometry, and also contains the fundamental theorem of plane geometry, including the theorem on the sum of the angles of a triangle and the Pythagorean theorem. In the 2nd book covers the fundamentals of geometric algebra. Third book is devoted to the properties of the circle, his tangents and chords. In the 4th book covers regular polygons, with the construction of a regular 15-gon belongs apparently to Euclid himself. Book 5th and 6th on the theory of relations and its application to solving algebraic problems. Book 7th, 8th and 9th on the theory of integers and rational numbers, the Pythagoreans developed no later than 5. BC. e. These three books are written, apparently, not on the basis of the extant works of Archytas.

The book discusses the 10th quadratic irrationality and presents the results obtained Theaetetus. In the book, the 11th covers the basics of stereochemistry. In the 12th book by using the method of exhaustion of Eudoxus proved theorems relating to the area of a circle and the volume of the ball, output ratio of the volumes of pyramids, cones, prisms and cylinders. The basis of the 13th book is based on the results obtained in the Theaetetus regular polyhedra. Books 14th and 15th Euclid do not belong, they were written later than the 14th - in the 2nd. BC and 15th - in 6.

## Other Facts

Second after the "Elements" of Euclid writing commonly called "Data" - an introduction to geometric analysis. Euclid also belongs to "Phenomena", devoted to the elementary spherical astronomy, "Optics " and " Catoptrics " little treatise "Sections of the canon " (contains ten problems of musical intervals), a collection of tasks by dividing the area of figures " about the division " (came to us in Arabic translation). Presentation in all these works, as in the "Elements", is subject to strict logic, the theorems are derived from precisely formulated physical hypotheses and mathematical postulates. Many works of Euclid are lost and their existence is in the past, we know only the links in the writings of other authors.

About the life of a scientist, almost nothing is known. The first commenter "Principia" Proclus (V century BC) could not specify when and where he was born and died Euclid. According to Proclus, "this pundit" lived during the reign of Ptolemy I. Some biographical data are preserved in the pages of the XII century Arabic manuscript: "Euclid, Naukrat son, known as "geometry", the old-time scholar, Greek in origin, place of residence of Syrian, originally from Tyre."

One legend says that King Ptolemy decided to study geometry. But it turned out that this is not so simple. Then he called and asked Euclid to show him an easy way to mathematics. " On the geometry is no royal road," - replied the scientist. So as the legend came to us, it became a popular expression.

King Ptolemy I, to exalt the state, the country has attracted scholars and poets, creating for them a temple of the Muses - Museion. There were rooms for classes, botanical and zoological gardens, astronomical study, astronomical tower, rooms for solitary work, and most importantly - a great library. Among the invited scientists proved and Euclid, which was founded in Alexandria - Egypt's capital - mathematical school and wrote for her students the fundamental work.

It was founded in Alexandria Euclid mathematical school and wrote a great work on geometry, combined under the title "Start" - the main work of his life. Believe that it was written about 325 BC.

Euclid's predecessors - Thales, Pythagoras, Aristotle and others have done a lot for the development of geometry. But all these were fragments, and not a single logic.

Both contemporaries and followers of Euclid attracted systematic and logical presentation of information. "Elements" consist of thirteen books, built on a single logical schema. Each of the thirteen books begins by defining the concepts of (point, line, plane, shape, etc.) which are used in it, and then based on a small number of key provisions (5 axioms and postulates 5) received no evidence construct the whole system geometry.

While the development of science and does not presuppose the practical methods of mathematics. Books I-IV cover geometry, the content goes back to the works of the Pythagorean School. In Book V developed the doctrine of proportions which is adjacent to Eudoxus of Cnidus. In books VII-IX contained the study of numbers representing the Pythagorean development of primary sources. In the books of X -XII contains the determination of areas in the plane and space (Geometry), the theory of irrationality (especially in book X); XIII in the book are correct research bodies, dating back to the Theaetetus.

"Elements" of Euclid is a summary of the geometry and is known today under the name of Euclidean geometry. She describes the metric properties of space that 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, infinite, isotropic, having three dimensions. Euclid gave a mathematical certainty atomistic idea of empty space in which atoms move. Simple geometrical object in Euclid’s definition is the point, which he defines as that which has no parts. In other words, the point - is indivisible atom space.

Infinity of space is characterized by three postulates: "From any point to any point, you can draw a straight line." "Limited direct can be continuously extended in a straight line." "From any center and a solution can be described as a circle."

Typically, the " Elements" of Euclid said that after the Bible is the most popular written by ancient monument. The book has a very remarkable history. For two thousand years, it was a handbook for students, used as the initial geometry course. "Elements" used extremely popular, and with them was shot multiple copies hardworking scribes in different cities and countries. Later "Elements" with papyrus parchment moved to, and then to the paper. For four centuries, "Elements" published in 2500 again, in the average yield of 6-7 publications annually. To the XX century book "Elements" was considered a basic textbook on geometry, not only for schools but also for universities.

"Elements" of Euclid were thoroughly studied by the Arabs and later European scientists. They were translated into major world languages . First originals were printed in 1533 in Basel is curious that the first translation into English, referring to 1570, was made by Henry Billingveem, London merchant

Euclid belong partly preserved, partly reconstructed in the future mathematical works he introduced an algorithm for the greatest common divisor of two integers randomly selected and algorithm called " account of Eratosthenes " - for finding prime numbers in a given number.

Euclid laid the foundations of geometrical optics, set them in the writings of "Optics " and " Catoptrics." The main concept of geometrical optics - straight light beam. Euclid argued that the light beam coming out of the eye (the theory of visual rays) as geometric constructions is not significant. He knows the law of reflection and focusing action of a concave spherical mirror, although the exact position of the focus cannot yet determine.

At Euclid we meet as a description of the monochord - a device to determine the pitch of the string and its parts. Believe that the monochord invented Pythagoras, Euclid and only described it ("Fission canon”, III century BC).

Euclid with his passion engaged computational system of interval relations. Monochord invention mattered for the development of music. Gradually a string of steel instead use two or three. Thus began the creation of keyboard instruments, first harpsichord, then piano. A root cause of the appearance of these musical instruments has become a mathematician.

Of course, all the features of the Euclidean space was opened immediately, but after centuries of scientific thought, but the starting point of this work were the "Elements" of Euclid. Basic knowledge of Euclidean geometry is now an essential element of general education throughout the world.

## Euclid and Ancient Philosophy

Since the time of the Pythagoreans and Plato arithmetic, music, geometry and astronomy (the so-called "Mathematical" sciences, later named as quadrivium) were considered as a model of systematic thinking and a preliminary stage for the study of philosophy. No accident occurred tradition according to which the entrance to Plato's Academy was the inscription: "Do not come in here who does not know geometry."

Geometric drawings, in which auxiliary lines during implicit truth becomes apparent to illustrate the doctrine of recollection, developed by Plato in Meno and other dialogues. Offers geometry therefore called theorems that their comprehension of truth requires the drawing is not easy to perceive the sensory impaired, but the "eyes of the mind." Everyone is drawing to a theorem is the idea that we see before us is this person, and conduct arguments and concludes once for all figures one with her kind.

Some "Platonism" Euclid is also associated with the fact that in the Timaeus Plato considered the doctrine of the four elements, which correspond to four regular polyhedra (tetrahedron - fire, octahedron - air, icosahedron - water cube - the land), the fifth is a polyhedron, a dodecahedron " became the inheritance of the figure of the universe. " In this regard, Beginnings can be regarded as detailed with all the necessary assumptions and ligaments doctrine of constructing five regular polyhedra - the so-called "Platonic solids", completes the proof of the fact that other regular solids, in addition to these five, does not exist.

For the Aristotelian doctrine of proof, developed in the second analyst, has also begun to provide rich material. Geometry at the beginning of construction as a lead system of knowledge in which all proposals are sequentially output one after the other along the chain, which is based on a small set of initial statements made without proof. According to Aristotle, such initial approval should be, because the chain output should start somewhere, so as not to be infinite. Further, trying to prove Euclid general allegations, this also corresponds to Aristotle’s favorite examples: «If every isosceles triangle has angles inherent in the amount equal to two right angles, then it is inherent in him not because he is an isosceles, but because he is a triangle” (An. Post? 85b12)?

## Works Cited

DeLacy, Estelle Allen (1963). Euclid and Geometry. New York: Franklin Watts.

Knorr, Wilbur Richard (1975). The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry. Dordrecht, Holland: D. Reidel. ISBN 90-277-0509-7.

Mueller, Ian (1981). Philosophy of Mathematics and Deductive Structure in Euclid's Elements. Cambridge, MA: MIT Press. ISBN 0-262-13163-3.

Reid, Constance (1963). A Long Way from Euclid. New York: Crowell.

Szabó, Árpád (1978). The Beginnings of Greek Mathematics. A.M. Ungar, trans. Dordrecht, Holland: D. Reidel. ISBN 90-277-0819-3.