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 billion years, this is only a short moment. During this “instant”, mankind made a huge leap from primitive stone tools to interplanetary spacecraft. It would be impossible if from time to time on the planet geniuses were not born that moved 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 his story.
How geometry came about
Since land allotments became the subject of sale and lease, their size and area had to be measured, including by calculation. In addition, such calculations became necessary in the construction of large-scale structures, as well as in measuring the volume of various objects. All this became the prerequisites for the emergence of 3-4 millennia ago in Egypt and Babylon, the art of land surveying. It was empirical and was a collection of examples of solutions to several hundred specific problems, without any evidence.
As a systematic science, geometry has developed in ancient Greece. By the third century BC, there was a large stock of facts and evidence-based methods. At the same time, the task arose to generalize the collected rather extensive geometric material. Hippocrates, Fedii, and other ancient Greek philosophers tried to solve it. However, a logically verified scientific system appeared only about 300 BC. e. with the publication of "Beginnings."
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 School of Mathematics. Almost 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 hometown, he founded a school where, together with his students, he created his famous work, which for more than two millennia has been the basis for any textbook on planimetry and stereometry.
"Beginnings" 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 ones deal with stereometry. As for the remaining volumes, they are devoted to arithmetic, which is given in terms of geometric postulates.
The role of the main work of Euclid in the subsequent development of the mathematical sciences is difficult to overestimate. We have heard several papyrus lists from the original, as well as Byzantine manuscripts.
In the Middle Ages, the "Beginnings" of Euclid were studied primarily by Arabs, who considered them one of the greatest works of human thought, and the scientist himself was a resident of Damascus. Much later, Europeans became interested in these works. With the advent of typography, science, including Euclidean geometry, has ceased to be the property of only a select few. After the first edition in 1533, the Beginnings became available to anyone who wanted to know the world, and there were more and more of them every year. Demand generated supply, therefore, it is believed that this work is the second among the most widely read monuments of antiquity after the Bible.
Some features
The Beginnings describes the metric properties of three-dimensional, empty, unlimited, and isotropic space, which is commonly called Euclidean. It is considered the arena where the phenomena of classical physics of Galileo and Newton take place.
An 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 Euclidean, it is he who determines the properties and geometry of Euclidean space.
According to scientists, the father of classical geometry created a perfect textbook, the study of which eliminates any misunderstanding of the material due to the way it is presented. In particular, each volume of the Beginning begins with the definition of concepts that are encountered for the first time. In particular, from the first pages of the 1st book, the reader learns what a point, line, straight line, etc. is. In total, there are 23 definitions needed to understand the main provisions of the material presented in this fundamental work.
Axioms and the first 4 postulates of Euclid
After the definitions, the author of "Beginnings" gives suggestions that are accepted without proof. He divides them into axioms and postulates. The first group consists of 11 statements that are known to a person intuitively. For example, the 8th axiom states that the whole is larger than the part, and according to the first, two quantities, separately equal to the third, are equal to each other.
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 with any radius it is possible to describe a circle;
- a bounded line can continue continuously in a straight line;
- all right angles are equal.
The fifth postulate of Euclid
For more than two millennia, this statement has repeatedly been the subject of close attention of mathematicians. However, we first get acquainted with the content of the fifth postulate of Euclid. So, in the modern formulation, it sounds like this: if on the plane at the intersection of two lines of the third the sum of the one-sided internal angles is less than 180 °, then these lines, when continued, sooner or later intersect from the side on which this quantity (sum) is less than 180 °.
The fifth postulate of Euclid, the formulation of which is given in different sources in different ways, from the very beginning aroused sport and the desire to transfer it to the category of theorems by constructing sound evidence. By the way, it is often replaced by another expression, actually coined by Proclus and also known as the Plaefer axiom. It says: on a plane through a point that does not belong to a given line, it is possible to draw one and only one straight line parallel to this.
Wording
As already mentioned, many scientists tried differently to express the idea of the 5th postulate of Euclid. Many formulations are fairly obvious. For instance:
- converging lines intersect;
- there is at least one rectangle, that is, a 4-gon with four right angles;
- each figure can be proportionally enlarged;
- there is a triangle having any arbitrarily large area.
disadvantages
Euclidean geometry became the greatest mathematical work of antiquity, and until the 19th century it reigned supreme in mathematics. Despite this, some of its shortcomings were noted by contemporaries of the author and ancient Greek scholars who lived a little 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 on the equality of right angles. A rigorous proof was found for him, thanks to which he went into the category of theorems.
History of the 5th postulate in antiquity and in the early Middle Ages
The classical formulation of this statement of Euclidean geometry seems much less obvious than the other four. It was this circumstance that haunted mathematicians.
The stumbling block for the fifth postulate of Euclid was the very definition of parallelism of two straight lines a and b, which states 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 made by the ancient Greek geometrical Posidonius. He suggested that the straight line parallel to the given set of all points of the plane that are at the same distance from the original. However, even this did not allow Posidonius to find proof of the 5th postulate.
Attempts by other mathematicians, including medieval ones, such as the Arabs of ibn Korr and Khayyam, did not lead to anything. The only thing that was achieved was 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 come close enough to the proof of the axiom of parallelism of Euclid. He wrote the outstanding textbook "The Beginnings of Geometry", which for about 150 years was the main one in teaching mathematics in schools of the Russian Empire. In it, the scientist presented three options for proving 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, independent of the fifth postulate, was given by the military engineer J. Boyai. But he himself was frightened of his discovery and did not begin to develop this idea, considering it erroneous. The great German mathematician C. Gauss was also unable to achieve success.
Breakthrough
For more than 2000 years, the fifth postulate of Euclid, whose proof hundreds of scientists tried to find, remained the number one problem in mathematics. The breakthrough was made by the Russian mathematician N. I. Lobachevsky. It was he who was the first in the world to describe the properties of real space, proving that the Euclidean geometry "works" only in the particular case of its system.
N. I. Lobachevsky initially went the same way as his colleagues. Trying to prove the 5th postulate, he did not succeed. Then the scientist refused the Euclidean view, according to which the sum of the angles of a triangle is 180 degrees. Then he began to prove this statement by contradiction and received a new formulation for the fifth postulate. Now he allowed the existence of several lines parallel to the given line and passing through a point lying outside this line.
New geometry
It makes no sense to discuss who has done more for mathematical science. The role of Euclid and Lobachevsky is comparable with the influence on the formation and development of physics of Newton and Einstein. At the same time, a new, absolute geometry made it possible to consider the concept of space, breaking away from the classical method of "I can only understand what I can measure." But it is precisely such an approach that has been practiced in science for many millennia.
Unfortunately, the ideas of Lobachevsky’s geometry were not accepted 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, cosmic infinity must be considered. Indeed, it is difficult to imagine that the boundless Universe is the sum of rectilinear spaces.
Lobachevsky geometry is used to describe curved spaces that are created by the gravitational fields of galaxies. It allowed us to move away from the method of reducing all figures to a “roughly regular” cylinder, circle, pyramid, or to an arbitrary combination of these figures. Indeed, for example, our planet in reality is not a ball, but a geoid, that is, a figure that is obtained by drawing the outer contour of the lithosphere (hard shell) of the Earth.
In real life, there are also analogues of curved spaces of the Universe, which allow us to imagine the possibility of the existence of several straight lines parallel to a given one passing through one point. In particular, these are curved surfaces of three types, which are distinguished by the Italian geometer E. Beltrami and called pseudospheres.
Further development of the Lobachevsky theory
The outstanding Russian was not the only one who suggested not the absoluteness of Euclidean geometry. In particular, the mathematician B. Riemann in 1854 put forward the idea of the possibility of the existence of spaces of zero, positive and negative curvature. This meant that it was possible to create an infinite number of different non-classical 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, through a point outside a given line it is impossible to draw a single line that is parallel to this one.
The situation is completely different with spaces of zero, negative, and positive curvature according to the theory of F. Klein. In particular, in the first case they are described by parabolic geometry, a particular case of which is classical, in the second they obey the ideas of Lobachevsky, and in the third they correspond to the properties described by Riemann.
After the publication of Albert Einstein's Theory of Relativity, ideas about such spaces were supplemented with data that takes into account the existence of four interdependent and changing dimensions - mass, energy, speed and time.
On practice
If we turn to the human perception of space, then within the earth's orbit for a giant triangle of the largest possible deviation of the sum of the internal angles from the classical 180 degrees will be only four millionths of a second. This value is beyond the scope of the possibilities of homo sapiens, therefore, for the “earthly” Euclidean geometry is in demand.
It remains to be expected when conditions are created that allow obtaining experimental data that confirm or refute the theories of N. Lobachevsky and B. Riemann on a galactic scale.
Now you know what the fifth postulate of Euclid declares and his story, which is very instructive and allows you to trace the evolution of human thought over the past 2300 years.