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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Euclidean geometry. History and essence of scientific discovery
Directory / The most important scientific discoveries Geometry, like other sciences, arose from the needs of practice. The very word "geometry" is Greek, in translation it means "surveying". People very early faced the need to measure land. This required a certain stock of geometric and arithmetic knowledge. Gradually, people began to measure and study the properties of more complex geometric shapes. “From the Egyptian papyri and ancient Babylonian texts that have come down to us, it can be seen that as early as 2 thousand years BC people were able to determine the areas of triangles, rectangles, trapezoids, and approximately calculate the area of a circle,” writes I. G. Bashmakova. “They also knew the formulas for determination of the volumes of a cube, a cylinder, a cone, a pyramid and a truncated pyramid.Information on geometry soon became necessary not only for measuring the earth.The development of architecture, and somewhat later astronomy, presented new requirements to geometry.In both Egypt and Babylon, colossal temples were built, the construction of which could be made only on the basis of preliminary calculations ... And yet, despite the fact that humanity has accumulated such an extensive knowledge of geometric facts, geometry as a science did not yet exist. Geometry became a science only after they began to systematically apply logical proofs in it, began to derive geometric sentences not only by direct measurements, but also by inference, by deriving one position from another, and establishing them in a general form. Usually this revolution in geometry is associated with the name of a scientist and philosopher of the XNUMXth century BC. Pythagoras of Samos". However, all new problems and theories created in connection with them led to the fact that the methods of mathematical proofs themselves improved, the need to create a coherent logical system in geometry increased. “But how to build such a system?” asks I.G. Bashmakova. “After all, we prove each individual sentence based on some other sentences. These sentences, in turn, are proved by reference to some three third sentences, etc., these references we could continue indefinitely, and the process of proof would never end. How to be? This circumstance was noticed in antiquity, and at the same time a way was found. Not later than the 2th century BC, Greek mathematicians, when constructing geometry, chose certain sentences that were accepted without proof, and all other proposals were deduced from them strictly logically. Proposals accepted without proof were called axioms and postulates. The most perfect example of such a theory for more than 300 thousand years was Euclid's Elements, written about XNUMX BC" . About life Euclid (circa 365 BC - 300 BC) almost nothing is known. Only a few legends about him have come down to us. The first commentator on the "Beginnings" Proclus (XNUMXth 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 are preserved on the pages of an Arabic manuscript of the XII century: Syrian, native of Tyre. One of the legends tells 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 way to mathematics. "There is no royal road to geometry," the scientist answered him. So, in the form of a legend, this expression, which has become popular, has come down to us. 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. 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. The predecessors of Euclid - Thales, Pythagoras, Aristotle and others did a lot for the development of geometry. But all these were separate fragments, not a single logical scheme. Both contemporaries and followers of Euclid were attracted by the systematic and logical nature of the information presented. "Beginnings" consists of 13 books built according to a single logical scheme. Each of the 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 of geometry is built . At that time, the development of science did not imply the existence 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 the Pythagorean primary sources. Books X-XII contain definitions of areas in the plane and space (stereometry), the theory of irrationality (especially in Book X); book XIII contains studies of regular bodies, going back to Theaetetus. Euclid's "Elements" is a presentation of that geometry, which is known to this day under the name of Euclidean geometry. As postulates, Euclid chose such sentences, which stated what can be verified by the simplest constructions using a compass and straightedge. Euclid also accepted some general axiom propositions, for example, that two quantities that are separately equal to a third are equal to each other. On the basis of such postulates and axioms, Euclid developed all planimetry strictly and systematically. In the Elements, he describes the metric properties of the space that modern science calls the 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. 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." "From every center and every solution a circle can be described." The doctrine of parallels and the famous fifth postulate ("If a line falling on two lines forms interior and on one side angles less than two lines, then these two lines extended indefinitely will meet on the side where the angles are less than two lines") define the properties of Euclidean space and its geometry, different from non-Euclidean geometries. It is usually said of the "Beginnings" that after the Bible it is the most popular written monument of antiquity. The book has a very interesting history. For two thousand years, it was a reference book for schoolchildren, used as an elementary course in geometry. The Elements were extremely popular, and many copies were made of them by industrious scribes in various cities and countries. Later, "Beginnings" moved from papyrus to parchment, and then to paper. Over the course of four centuries, the "Principles" were published 2500 times: on average, 6-7 editions were published annually. Until the twentieth century, the book was considered the main textbook on geometry, not only for schools, but also for universities. The "beginnings" of Euclid were thoroughly studied by the Arabs, and later by European scientists. They have been translated into the main world languages. The first originals were printed in 1533 in Basel. Curiously, the first translation into English, dating back to 1570, was made by Henry Billingway, a London merchant. Of course, all the features of the Euclidean space were not discovered immediately, but as a result of the centuries-old work of scientific thought, but the starting point of this work was the "Beginnings" of Euclid. Knowledge of the foundations of Euclidean geometry is now a necessary element of general education throughout the world. We can safely say that Euclid laid the foundations not only of geometry, but of all ancient mathematics. Only in the nineteenth century did the study of the foundations of geometry rise to a new, higher level. It was possible to find out that Euclid did not list all the axioms that are actually needed to construct geometry. In fact, the scientist used them in the proofs, but did not formulate them. Nevertheless, all of the above does not in the least detract from the role of Euclid, who was the first to show how it is possible and how to build a mathematical theory. He created the deductive method, firmly established in mathematics. This means that all subsequent mathematicians are, to a certain extent, students of Euclid. Author: Samin D.K.
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Comments on the article: Fedor Aggeev That's right, with the amendment that we need to talk more and more on this topic. Sincerely, Fedor Aggeev.
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