The fifth postulate of Euclid: formulation

It is believed that the first human civilizations appeared 10,000 years ago. Compared with the age of our planet, which, according to scientists, is about 4.54 million years old, this is only a brief moment. For this "moment" humanity has made a huge leap from primitive stone tools to interplanetary spaceships. It would be impossible if from time to time on the planet geniuses would not be born, moving science forward. Among them, of course, is Euclid. His works became the basis and powerful impetus for the development of modern mathematics.

This article is devoted to the fifth postulate of Euclid and its history.

How Geometry

Since the land plots became the subject of sale and lease, their size and area had to be measured, including by calculations. In addition, such calculations became necessary for the construction of large-scale structures, as well as for measuring the volume of various items. All this became the prerequisites for the emergence of the art of land surveying in Egypt and Babylonia 3-4 millennia ago. It was empirical and represented a collection of examples of solving several hundred specific problems, without any evidence.

As a systematic science, geometry developed in ancient Greece. By the third century BC there was a large stock of facts and evidence. At the same time, the task arose to generalize the collected rather extensive geometric material. It tried to solve Hippocrates, Fediy and other ancient Greek philosophers. However, a logically adjusted scientific system appeared only about 300 BC. E. With the publication of "Elements".

Who was Euclid

Ancient Greece gave the world many of the greatest philosophers and scientists. One of them is Euclid, who became the founder of the Alexandria Mathematical School. Practically nothing is known about the scientist himself. Some sources indicate that in his youth the future Father of modern geometry studied at the famous school of Plato in Athens, and then returned to Alexandria, where he continued to study mathematics and optics, and also wrote music. In his native city, he founded a school where, together with his students, he created his famous work, which for more than two millennia is the basis for any textbook on planimetry and stereometry.

"The Beginning" of Euclid

The main and first most systematic work on geometry consists of 13 volumes. The first four and sixth books deal with planimetry, and the 11th, 12th and 13th are stereometric. As for the remaining volumes, they are devoted to arithmetic, which is given in terms of geometric postulates.

The role of Euclid's main work in the subsequent development of mathematical sciences can not be overestimated. Several papyrus lists from the original, as well as Byzantine manuscripts, have reached us.

In the Middle Ages, the "Elements" of Euclid were studied primarily by the Arabs, who considered them one of the greatest works of human thought, and the scientist himself a resident of Damascus. Much later these works interested the Europeans. With the advent of book printing, science, including the geometry of Euclid, ceased to be the property of only the elect. After the first edition in 1533, the "Elements" became available to everyone who wanted to know the world, and every year it became more and more. Demand spawned the proposal, so it is believed that this work is the second among the most widely read ancient sites after the Bible.

Some features

The "Beginnings" describe the metric properties of a three-dimensional, empty, infinite and isotropic space, which is commonly called Euclidean. It is considered an arena, where the phenomena of classical physics of Galileo and Newton occur.

Elementary geometric object, according to Euclid, is a point. The second important concept is the infinity of space, which is characterized by the first three postulates. The fourth concerns the equality of right angles. As for the fifth postulate of Euclid, it is he who determines the properties and geometry of Euclidean space.

According to scientists, the father of classical geometry has created a perfect textbook, in the study of which any misunderstanding of the material is excluded due to the way it is presented. In particular, each volume of "Beginnings" begins with the definition of the concepts encountered for the first time. In particular, from the first pages of the first book, the reader will learn what a point, line, line, etc. All in all there are 23 definitions necessary to understand the main points of the material presented in this fundamental work.

Axioms and the first four postulates of Euclid

After the definitions, the author of "Nachal" quotes proposals that are accepted without proof. They divide them into axioms and postulates. The first group consists of 11 statements that are intuitively known to a person. For example, the 8th axiom states that the whole is greater than the part, and according to the first, two quantities that are separately equal to the third are equal.

In addition, Euclid gives 5 postulates. The first four read:

• From any point to any other one can draw a straight line;
• From any center of any radius it is possible to describe a circle;
• The bounded line can proceed continuously along a straight line;
• All right angles are equal.

The fifth postulate of Euclid

For more than two millennia this statement has repeatedly become the object of close attention of mathematicians. However, first let us become acquainted with the content of the fifth postulate of Euclid. So, in the modern formulation, it sounds like this: if on the plane with the intersection of two straight lines the third sum of one-side internal angles is less than 180 °, then these straight lines will intersect sooner or later on the side with which this value (sum) is less than 180 °.

The fifth postulate of Euclid, the formulation of which in different sources is given differently, from the very beginning caused the sport and the desire to translate it into the category of theorems by building a well-grounded proof. By the way, it is often replaced by another expression, actually invented by Proclus and known also as the axiom of Playfair. It says: on the plane through a point not belonging to a given line, it is possible to draw one and only one straight line parallel to this one.

Formulations

As already mentioned, many scientists have tried differently to express the idea of the 5th postulate of Euclid. Many formulations are quite obvious. For example:

• The approaching straight lines intersect;
• There exists at least one rectangle, that is, a 4-gon with four right angles;
• Each figure can be proportionally increased;
• There exists a triangle having any area of any size that is arbitrarily large.

disadvantages

The geometry of Euclid became the greatest mathematical work of antiquity and up to the 19th century it reigned supreme in mathematics. Despite this, some of its shortcomings were noted by the author's contemporaries and ancient Greek scholars who lived somewhat later. In particular, Archimedes added a new axiom, named after him. It says: for any segments AB and CD there exists a natural number n such that n · [AB]> [CD].

In addition, scientists sought to minimize the system of Euclidean postulates and axioms. To do this, they brought some of them out of the rest.

So it was possible to "get rid" of the 4th postulate about the equality of right angles. For him, a rigorous proof was found, which made him a theorist.

History of the 5th postulate in antiquity and in the early Middle Ages

The classical formulation of this statement of Euclid's geometry seems much less obvious than the other four. It was this circumstance that did not bother mathematicians.

The stumbling block for the fifth postulate of Euclid was the very definition of the parallelism of two straight lines a and b, which says that the sum of two one-sided angles, which are formed by the intersection of a and b with the third straight line c, is 180 degrees.

The first attempt to prove it as a theorem was undertaken by the ancient Greek geometer Posidonius. He proposed that the set of all points on the plane that are at the same distance from the original plane be considered as a direct parallel to the given one. However, even this did not allow Posidonia to find evidence of the 5th postulate.

The attempts of other mathematicians, including medieval ones, such as the Arabs of Ibn Korra and Hayam, did not lead to anything. The only thing that has been achieved is the emergence of new postulates, which are proved taking into account various assumptions.

In the 18-19th centuries

Classical geometry continued to interest mathematicians in the 18th century. In particular, the French mathematician A. Legendre was able to approach quite close to the proof of the Euclidean parallelism axiom. His pen belongs to the outstanding textbook "The Beginnings of Geometry", which for about 150 years was the main one for teaching mathematics in the schools of the Russian Empire. In it, the scientist gave three variants of the proof of the Euclidean axiom of parallelism, but all of them turned out to be incorrect.

By the beginning of the 19th century, the idea of creating non-Euclidean geometry arose. The first description of the system, which does not depend on the fifth postulate, was given by the military engineer J. Boyay. But he himself was frightened of his discovery and did not develop this idea, considering it erroneous. The great German mathematician K. Gauss was also unable to achieve success.

Breakthrough

For more than 2000 years, the fifth postulate of Euclid, the proof of which hundreds of scientists tried to find, remained the number one problem in mathematics. The breakthrough was made by the Russian mathematician NI Lobachevsky. He was the first in the world to describe the properties of real space, proving that Euclid's geometry "works" only in the particular case of his system.

NI Lobachevsky initially followed the same path as his colleagues. Trying to prove the 5th postulate, he did not succeed. Then the scientist abandoned the Euclidean notion, according to which the sum of the angles of the triangle is 180 degrees. Further, he began to prove this assertion from the contrary and received a new formulation for the fifth postulate. Now he allowed the existence of several lines parallel to a given one, and passing through a point lying outside this line.

New geometry

There is no point in discussing who did more for mathematical science. The role of Euclid and Lobachevsky is comparable with the influence on the formation and development of the physics of Newton and Einstein. At the same time, the new, absolute geometry allowed us to consider the concept of space, detached from the classical method "I can only understand what I can measure." But this is the approach that has been practiced in science for many millennia.

Unfortunately, the ideas of Lobachevsky's geometry were not perceived and understood by contemporaries. In particular, his students did not continue the work of the scientist, and the development of non-Euclidean geometry was postponed for several decades.

Some features of the Lobachevsky theory

To understand the new geometry, we need to consider cosmic infinity. Indeed, it is difficult to imagine that the boundless universe is a sum of rectilinear spaces.

The geometry of Lobachevsky is used to describe the curvilinear spaces that are created by the gravitational fields of galaxies. She allowed to move away from the method of reducing all figures to an "approximately right" cylinder, circle, pyramid, or an arbitrary combination of these figures. After all, for example, our planet in reality is not a sphere, but a geoid, that is, a figure that is obtained by outlining the outer contour of the Earth's lithosphere (solid shell).

In real life, there are analogues of the curvilinear spaces of the Universe, which allow one to imagine the possibility of the existence of several direct parallel ones passing through one point. In particular, these are curved surfaces of three types, which are distinguished by the Italian geometry E. Beltrami and called pseudospheres.

Further development of Lobachevsky's theory

The outstanding Russian was not the only one who suggested that the Euclidean geometry is not absolute. In particular, the mathematician B. Riemann in 1854 advanced the idea of the possibility of the existence of spaces of zero, positive and negative curvature. This meant that it is possible to create an infinite number of different nonclassical geometries.

From the position of B. Riemann, who studied mainly spaces with positive curvature, the 5th postulate of Euclid sounds quite unexpectedly. According to his ideas, no straight line can be drawn through a point outside this line, which is parallel to the given one.

The situation is completely different with spaces of zero, negative, and positive curvature according to Klein's theory. In particular, in the first case they are described by parabolic geometry, the special case of which is classical, in the second case they obey the ideas of Lobachevsky, and in the third they correspond to the properties described by Riemann.

After the publication of the Relativity Theory of Albert Einstein, the concepts of such spaces were supplemented by data that took into account the existence of four interdependent and changing dimensions - mass, energy, velocity and time.

On practice

If we go to the human perception of space, then within the terrestrial orbit for the giant triangle, the largest possible deviation of the sum of internal angles from the classical 180 degrees is only four millionths of a second. Such a value is beyond the capabilities of homo sapiens, so Euclid's geometry is in demand for "earthly" ones.

It remains to wait for conditions to be created that make it possible to obtain experimental data that confirm or disprove the theories of N. Lobachevsky and B. Riemann in the scale of the Galaxy.

Now you know that declares the fifth postulate of Euclid and its history, which is very instructive and allows you to trace the evolution of human thought over the past 2300 years.