Euclid - Студенческий научный форум

XI Международная студенческая научная конференция Студенческий научный форум - 2019

Euclid

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Ancient Greek thinker Euclid became the first mathematician of the Alexandrian school and the author of one of the most ancient theoretical mathematical treatises. The biography of this scientist is known much less than about his work. Thus, in the well-known work of "The Beginning" Euclid outlined the stereometry, planimetry, aspects of number theory, created the basis for the subsequent development of mathematics. Biography Euclid supposedly began in 325 BC (this is an approximate date, the exact year of birth is unknown) in Alexandria. Some researchers suggest that the future mathematician was born in Tire, and spent most of his adult life in Damascus. Probably, Euclid came from a rich family, since he studied at the Athenian school of Plato (at that time such education was available only to wealthy citizens). The scientists managed to establish that the author of "Elements" was younger than the famous followers of Plato, who lived and worked in the period from 427 to 347 centuries BC, but older than Archimedes, who was born in 287 and died in 212 BC. Euclid understood the philosophical concept of Plato and shared her basic theses. The above information about the personality and life path of Euclid is drawn by researchers from Proclus's comments, which he wrote to the first book of the "Beginning." The statements of Stobei and Pappa about the personality of the ancient Greek thinker are also known. Stobey allegedly told that in reply to the student's question about the benefits of science, Euclid ordered the slave to give him some coins. Papp also noted that the scientist was able to be gracious and gentle with any person who could at least to some extent be useful for the development of mathematical sciences. The surviving data on Euclid are so small and doubtful that there was a version about the appropriation of the pseudonym "Euclid" to entire collectives of scientists from ancient Alexandria. Euclid of Alexandria is confused with the Greek philosopher Euclid of Megar, a disciple of Socrates, who lived in the 400th century BC. In the Middle Ages, Euclid from Megar was even considered the author of The Beginnings. The surviving data on Euclid are so small and doubtful that there was a version about the appropriation of the pseudonym "Euclid" to entire collectives of scientists from ancient Alexandria. Euclid of Alexandria is confused with the Greek philosopher Euclid of Megar, a disciple of Socrates, who lived in the 400th century BC. In the Middle Ages, Euclid from Megar was even considered the author of The Beginnings.

Mathematics

Euclid spent much of his free time at the Alexandria Library, a temple of knowledge founded by Ptolemy. Within the walls of this institution, the ancient Greek scientist began to integrate arithmetic laws, geometric principles and the theory of irrational numbers into geometry. Euclid described the results of his works in the book "Elements" - a work that brought a great contribution to the development of mathematics. The book consists of fifteen volumes:

In the book I the author tells about the properties of parallelograms and triangles, finishing the presentation using the Pythagorean theorem when calculating the parameters of rectangular triangles.

Book number II describes the principles and patterns of geometric algebra and goes back to the luggage of knowledge accumulated by the Pythagoreans.

In books III and IV, Euclid considers the geometry of circles described and inscribed in polygons. During the creation of these volumes, the author could turn to the use of the works of Hippocrates of Chios.

In the fifth book, the Greek mathematician considered the general theory of proportions developed by Eudoxus of Cnidus.

In the materials of the VI book the author applies the general theory of Eudoxus' proportions to the theory of similar figures.

Books numbered VII-IX describe the theory of numbers. When writing these volumes, the mathematician again turned to the materials created and assembled by the Pythagoreans - representatives of the teaching, in which the number plays a central role. In these works the author speaks about geometric progressions and proportions, proves the infinity of the set of prime numbers, studies even perfect numbers, introduces the concept of GCD (the greatest common divisor). The algorithm for finding such a divisor is now called the Euclidean algorithm. There is a suggestion that the Eighth book was written not by Euclid himself, but by Archite of Tarentum.

Volume number X is the most complex and voluminous work in the "Elements", which contains the classification of irrationalities. The authorship of this book is also not known for certain: it could be written both by Euclid himself and by the Teathet Athenian.

On pages XI of the book the mathematician tells about the fundamentals of stereometry.

Book XII contains proofs of theorems on the volumes of cones and pyramids, the ratio of the areas of circles. To construct these proofs, the exhaustion method is used. Most researchers agree that this book was not written by Euclid. The likely author is Eudoxus of Cnidus.

Materials of the XIII book contain information on the construction of five regular polyhedra ("Platonic solids"). Some part of the constructions given in the volume could be developed by the Teethet Athenian.

Books XIV and XV, according to generally recognized opinion, also belong to other authors. So, the penultimate volume of the "Beginnings" was written by Gipsikl (also living in Alexandria, but later Euclid), and the last one was Isidore of Miletus (who built the temple of St. Sophia in Constantinople in the beginning of the sixth century BC).

Before the appearance of the "Elements" of Euclid, works of the same name, the essence of which consisted in the sequential presentation of the key facts of theoretical arithmetic and geometry, were compiled by Leont, Hippocrates of Chios, and Feb.. All of them have practically disappeared from use after the appearance of Euclid's work.

For two thousand years, fifteen volumes of "Elements" acted as a basic textbook on geometry. The work is translated into Arabic, then into English. The "beginnings" were reprinted hundreds of times, and the basic mathematical calculations mentioned in them remain relevant to this day. A considerable part of the materials that the author has included in his work are not his own discoveries, but the theories known earlier. The essence of Euclid's work consisted in the processing of material, its systematization and the reduction of disparate data together. Some books Euclid began with a list of definitions, in the first book there is also a list of axioms and postulates.

The postulates of Euclid are divided into two groups: general concepts, including generally accepted scientific statements, and geometric axioms. So, in the first group there are such statements:

"If two values ​​are separately equal to the same third, then they are equal to each other."

"The whole is greater than the sum of parts."

In the second group there are, for example, the following statements:

"From every point to every point, you can draw a straight line."

"All right angles are equal to each other".

"Beginning" is not the only book written by Euclid. He also wrote a number of works on katoptryke (a new branch of optics, in no small measure asserting the mathematical function of mirrors). Several studies the scientist devoted to the study of conical sections. The mathematician also developed assumptions and hypotheses concerning the trajectory of motion of bodies and the laws of mechanics. He became the author of the key tools that geometry deals with - the so-called "Euclidean constructions". Many works of this ancient Greek thinker have not reached our days. Among many other mathematical gems, the thirteen volumes of the “Elements” contain formulas for calculating the volumes of solids such as cones, pyramids and cylinders; proofs about geometric series, perfect numbers and primes; algorithms for finding the greatest common divisor and least common multiple of two numbers; a proof and generalization of Pythagoras’ Theorem, and proof that there are an infinite number of Pythagorean Triples; and a final definitive proof that there can be only five possible regular Platonic Solids.

However, the “Elements” also includes a series of theorems on the properties of numbers and integers, marking the first real beginnings of number theory. For example, Euclid proved what has become known as the Fundamental Theorem of Arithmethic (or the Unique Factorization Theorem), that every positive integer greater than 1 can be written as a product of prime numbers (or is itself a prime number). Thus, for example: 21 = 3 x 7; 113 = 1 x 113; 1,200 = 2 x 2 x 2 x 2 x 3 x 5 x 5; 6,936 = 2 x 2 x 2 x 3 x 17 x 17; etc. His proof was the first known example of a proof by contradiction (where any counter-example, which would otherwise prove an idea false, is shown to makes no logical sense itself).

He was the first to realize - and prove - that there are infinitely many prime numbers. The basis of his proof, often known as Euclid’s Theorem, is that, for any given (finite) set of primes, if you multiply all of them together and then add one, then a new prime has been added to the set (for example, 2 x 3 x 5 = 30, and 30 + 1 = 31, a prime number) a process which can be repeated indefinitely.

Euclid also identified the first four “perfect numbers”, numbers that are the sum of all their divisors (excluding the number itself):
    6 = 1 + 2 + 3;
    28 = 1 + 2 + 4 + 7 + 14;
    496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248; and
    8,128 = 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064.
He noted that these numbers also have many other interesting properties. For example:

They are triangular numbers, and therefore the sum of all the consecutive numbers up to their largest prime factor: 6 = 1 + 2 + 3; 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7; 496 = 1 + 2 + 3 + 4 + 5 + .... + 30 + 31; 8,128 = 1 + 2 + 3 + 4 + 5 + ... + 126 + 127. Their largest prime factor is a power of 2 less one, and the number is always a product of this number and the previous power of two: 6 = 21(22 - 1); 28 = 22(23 - 1); 496 = 24(25 - 1); 8,128 = 26(27 - 1).

Although the Pythagoreans may have been aware of the Golden Ratio (φ, approximately equal to 1.618), Euclid was the first to define it in terms of ratios (AB:AC = AC:CB), and demonstrated its appearance within many geometric shapes.

Euclid's Axioms

Euclid based his approach upon 10 axioms, statements that could be accepted as truths. He called these axioms his 'postulates' and divided them into two groups of five, the first set common to all mathematics, the second specific to geometry. Some of these postulates seem to be self-explanatory to us, but Euclid operated upon the principle that no axiom could be accepted without proof.

Euclid's First Group of Postulates - the Common Notions:

Things which are equal to the same thing are also equal to each other

If equals are added to equals, the results are equal

If equals are subtracted from equals, the remainders are equal

Things that coincide with each other are equal to each other

The whole is greater than the part

The remaining five postulates were related specifically to geometry:

A straight line can be drawn between any two points.

Any finite straight line can be extended indefinitely in a straight line.

For any line segment, it is possible to draw a circle using the segment as the radius and one end point as the center.

All right angles are congruent (the same).

If a straight line falling across two other straight lines results in the sum of the angles on the same side less than two right angles, then the two straight lines, if extended indefinitely, meet on the same side as the side where the angle sums are less than two right angles. Euclid felt that anybody who could read and understand words could understand his notions and postulates but, to make sure, he included 23 definitions of common words, such as 'point' and 'line', to ensure that there could be no semantic errors. From this basis, he built his entire theory of plane geometry, which has shaped mathematics, science and philosophy for centuries. He proved that it is an impossibility to find the 'largest prime number,' because taking the largest known prime number and adding one to the product of all previous primes and the largest prime will give you another, larger prime number.

Philosophy

In ancient times, philosophy was closely intertwined with many other branches of scientific knowledge. So, geometry, astronomy, arithmetic and music were considered mathematical sciences, understanding of which is necessary for a qualitative study of philosophy. Euclid developed Plato's doctrine of the four elements, which are associated with four regular polyhedra:

the elements of fire represent the tetrahedron;

The air element corresponds to an octahedron;

the element of the earth is associated with the cube;

the water element communicates with the icosahedron.

In this context, the "Principles" can be regarded as a kind of teaching on the construction of "Platonic solids", that is, five regular polyhedra. The teaching contains all the necessary prerequisites, proofs and bundles. The proof of the possibility of constructing such bodies is completed by affirming the fact that no other regular bodies, with the exception of the given five, exist.

Practically every theorem of Euclid in the "Beginnings" corresponds also to the indicators of the doctrine of Aristotle's proof. So, the author consistently deduces the consequences from the causes, forming a chain of logical evidence. Moreover, he proves even statements of a general nature, which also corresponds to the teachings of Aristotle.

Euclid's Influence

The reason that Euclid was so influential is that his work is more than just an explanation of geometry or even of mathematics. The way in which he used logic and demanded proof for every theorem shaped the ideas of western philosophers right up until the present day. Great philosopher mathematicians such as Descartes and Newton presented their philosophical works using Euclid's structure and format, moving from simple first principles to complicated concepts. Abraham Lincoln was a fan, and the US Declaration of Independence used Euclid's axiomatic system.

Apart from the Elements, Euclid also wrote works about astronomy, mirrors, optics, perspective and music theory, although many of his works are lost to posterity. Certainly, he can go down in history as one of the greatest mathematicians of all time, and he was certainly one of the giants upon whose shoulders Newton stood.

Personal life

We have received only some information about the work of Euclid in science, about his personal life, almost nothing is known. There is a legend that King Ptolemy, who decided to study geometry, was annoyed by its complexity. Then he turned to Euclid and asked him to point to an easier way to knowledge, to which the thinker replied: "There is no royal road to geometry." The expression subsequently became winged. There is evidence that under the Alexandria Library this ancient Greek scholar founded a private mathematical school. It was studied by the same enthusiasts of science as Euclid himself. Even at the end of his life, Euclid helped his students write works, create their own theories and develop relevant proofs.

There is no precise data on the appearance of the scientist. His portraits and sculptures are the fruit of the imagination of their creators, a fictitious image passed down from generation to generation.

Death

Presumably, Euclid died in the 260th years BC. The exact causes of death are not known. The scientist's legacy survived him for two thousand years and inspired many great people centuries after his death.

There is an opinion that politician Abraham Lincoln liked to quote Euclid's statements in his speeches and had several volumes of "Elements" with him.

The scientists of later years based the works on the works of Euclid. Thus, the Russian mathematician Nikolai Lobachevsky used the materials of an ancient Greek thinker to develop hyperbolic geometry, or Lobachevsky's geometry. The mathematics format that Euclid created was now known as "Euclidean geometry." The scientist also created a device for determining the pitch of the string and studied the interval relationships, contributing to the creation of keyboard musical instruments.

To put it briefly Euclid of Alexandria (not to be confused with Socrates' student, Euclid of Megara, who lived a century earlier), directed the school of mathematics at the great university of Alexandria. Little else is known for certain about his life, but several very important mathematical achievements are credited to him. He was the first to prove that there are infinitely many prime numbers; he produced an incomplete proof of the Unique Factorization Theorem (Fundamental Theorem of Arithmetic); and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect (in the sense of Pythagoras) if M is Mersenne. (The converse, that any even perfect number has such a corresponding Mersenne prime, was tackled by Alhazen and proven by Euler.) His books contain many famous theorems, though much of the Elements was due to predecessors like Pythagoras (most of Books I and II), Hippocrates (Book III), Theodorus, Eudoxus (Book V), Archytas (perhaps Book VIII) and Theaetetus. Book I starts with an elegant proof that rigid-compass constructions can be implemented with a collapsing compass. (Given A, B, C, find CF = AB by first constructing equilateral triangle ACD; then use the compass to find E on AD with AE = AB; and finally find F on DC with DF = DE.) Although notions of trigonometry were not in use, Euclid's theorems include some closely related to the Laws of Sines and Cosines. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Euclid ranks #14 on Michael Hart's famous list of the Most Influential Persons in History. The Elements introduced the notions of axiom and theorem; was used as a textbook for 2000 years; and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.

There are many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant and "went home to my father's house [to read Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."

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