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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Pythagorean theorem. History and essence of scientific discovery
Directory / The most important scientific discoveries It's hard to find a person with a name Pythagoras would not be associated with the Pythagorean theorem. Even those who are far from mathematics in their lives continue to remember the "Pythagorean pants" - a square on the hypotenuse, equal in size to two squares on the legs. The reason for such popularity of the Pythagorean theorem is clear: it is simplicity - beauty - significance. Indeed, the Pythagorean theorem is simple, but not obvious. The contradiction of the two principles gives it a special attractive force, makes it beautiful. But, in addition, the Pythagorean theorem is of great importance. It is used in geometry literally at every step. There are about five hundred different proofs of this theorem, which indicates a gigantic number of its specific implementations. Historical studies date the birth of Pythagoras around 580 BC. Happy father Mnesarchus surrounds the boy with cares. He had the opportunity to give his son a good upbringing and education. The future great mathematician and philosopher already in childhood showed great abilities for the sciences. From his first teacher, Hermodamas, Pythagoras receives knowledge of the basics of music and painting. For memory exercises, Hermodamas forced him to learn songs from the Odyssey and the Iliad. The first teacher instilled in young Pythagoras a love for nature and its mysteries. Several years have passed, and on the advice of his teacher, Pythagoras decides to continue his education in Egypt. With the help of a teacher, Pythagoras manages to leave the island of Samos. But while Egypt is far away. He lives on the island of Lesbos with his relative Zoilus. There, Pythagoras meets the philosopher Ferekid, a friend of Thales of Miletus. Pythagoras studied astrology, prediction of eclipses, the secrets of numbers, medicine and other sciences obligatory for that time from Pherekides. Then, in Miletus, he listens to the lectures of Thales and his younger colleague and student Anaximander, an eminent geographer and astronomer. Pythagoras acquired a lot of important knowledge during his stay at the Milesian school. Before Egypt, he stops for a while in Phoenicia, where, according to legend, he studies with the famous Sidonian priests. Studying Pythagoras in Egypt contributes to the fact that he became one of the most educated people of his time. Here Pythagoras falls into Persian captivity. According to ancient legends, in captivity in Babylon, Pythagoras met with Persian magicians, joined Eastern astrology and mysticism, and became acquainted with the teachings of the Chaldean sages. The Chaldeans introduced Pythagoras to the knowledge accumulated by the Eastern peoples over many centuries: astronomy and astrology, medicine and arithmetic. Pythagoras spent twelve years in Babylonian captivity until he was released by the Persian king Darius Hystaspes, who heard about the famous Greek. Pythagoras is already sixty, he decides to return to his homeland in order to introduce his people to the accumulated knowledge. Since Pythagoras left Greece, there have been great changes. The best minds, fleeing the Persian yoke, moved to Southern Italy, which was then called Great Greece, and founded the colony cities of Syracuse, Agrigent, Croton there. Here Pythagoras is planning to create his own philosophical school. Pretty quickly, he is gaining great popularity among the residents. Pythagoras skillfully uses the knowledge gained in wandering around the world. Over time, the scientist stops speaking in temples and on the streets. Already in his home, Pythagoras taught medicine, the principles of political activity, astronomy, mathematics, music, ethics and much more. Outstanding political and statesmen, historians, mathematicians and astronomers came out of his school. It was not only a teacher, but also a researcher. His students also became researchers. Pythagoras developed the theory of music and acoustics, creating the famous "Pythagorean scale" and conducting fundamental experiments on the study of musical tones: he expressed the ratios found in the language of mathematics. In the School of Pythagoras, for the first time, a conjecture was made about the sphericity of the Earth. The idea that the movement of celestial bodies is subject to certain mathematical relationships, the ideas of "harmony of the world" and "music of the spheres", which subsequently led to a revolution in astronomy, first appeared precisely in the School of Pythagoras. The scientist also did a lot in geometry. Proclus assessed the contribution of the Greek scientist to geometry as follows: "Pythagoras transformed geometry, giving it the form of a free science, considering its principles in a purely abstract way and exploring theorems from an immaterial, intellectual point of view. It was he who found the theory of irrational quantities and the construction of cosmic bodies." In the school of Pythagoras, geometry for the first time takes shape as an independent scientific discipline. It was Pythagoras and his students who were the first to study geometry systematically - as a theoretical doctrine of the properties of abstract geometric figures, and not as a collection of applied recipes for land surveying. The most important scientific merit of Pythagoras is the systematic introduction of proof into mathematics, and, above all, into geometry. Strictly speaking, only from this moment does mathematics begin to exist as a science, and not as a collection of ancient Egyptian and ancient Babylonian practical recipes. With the birth of mathematics, science in general is also born, for "no human research can be called true science if it has not gone through mathematical proofs" (Leonardo da Vinci). So, the merit of Pythagoras was that he, apparently, was the first to come to the following idea: in geometry, firstly, abstract ideal objects should be considered, and, secondly, the properties of these ideal objects should not be established from using measurements on a finite number of objects, but using reasoning that is valid for an infinite number of objects. This chain of reasoning, which, with the help of the laws of logic, reduces non-obvious statements to known or obvious truths, is a mathematical proof. The discovery of the theorem by Pythagoras is surrounded by a halo of beautiful legends. Proclus, commenting on the last sentence of book 1 of "Beginnings" Euclid, writes: "If you listen to those who like to repeat ancient legends, you will have to say that this theorem goes back to Pythagoras; they say that in honor of this discovery he sacrificed a bull." However, more generous storytellers turned one bull into one hecatomb, and this is already a whole hundred. And although Cicero also noted that any shedding of blood was alien to the charter of the Pythagorean order, this legend firmly merged with the Pythagorean theorem and continued to evoke warm responses two thousand years later. Mikhail Lomonosov on this occasion, he wrote: "Pythagoras sacrificed a hundred oxen to Zeus for the invention of one geometric rule. But if for the rules found in modern times from witty mathematicians to act according to his superstitious jealousy, then it would hardly be possible to find so many cattle in the whole world." A.V. Voloshinov in his book about Pythagoras notes: "And although today the Pythagorean theorem is found in various particular problems and drawings: in the Egyptian triangle in the papyrus of the time of Pharaoh Amenemhet I (about 2000 BC), and in the Babylonian cuneiform tablets era of King Hammurabi (XVIII century BC), and in the ancient Chinese treatise "Zhou-bi suan jin" ("Mathematical treatise on the gnomon"), the time of creation of which is not exactly known, but where it is stated that in the XII century BC the Chinese knew the properties of the Egyptian triangle, and by the XNUMXth century BC - and the general form of the theorem, and in the ancient Indian geometric and theological treatise of the XNUMXth-XNUMXth centuries BC "Sulva Sutra" ("Rules of the Rope"), - despite all this, the name of Pythagoras is so firmly fused with the Pythagorean theorem that it is simply impossible to imagine that this phrase will fall apart. Today it is generally accepted that Pythagoras gave the first proof of the theorem bearing his name. Alas, no trace of this evidence has survived either. Therefore, we have no choice but to consider some of the classical proofs of the Pythagorean theorem, known from ancient treatises. It is also useful to do this because modern school textbooks give an algebraic proof of the theorem. At the same time, the primordial geometric aura of the theorem disappears without a trace, that thread of Ariadne that led the ancient sages to the truth is lost, and this path almost always turned out to be the shortest and always beautiful. The Pythagorean theorem states: "The square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs." The simplest proof of the theorem is obtained in the simplest case of an isosceles right triangle. Probably, the theorem began with him. Indeed, it is enough just to look at the tiling of isosceles right triangles to see that the theorem is true. In the 4nd century BC, paper was invented in China and at the same time the creation of ancient books began. This is how "Mathematics in Nine Books" appeared - the most important of the surviving mathematical and astronomical works. In the IX book of "Mathematics" there is a drawing proving the Pythagorean theorem. The key to this proof is not difficult to find. Indeed, in the ancient Chinese drawing there are four equal right-angled triangles with legs and a hypotenuse. C are stacked so that their outer contour forms a square with side A + B, and the inner one - a square with side C, built on the hypotenuse. If a square with side c is cut out and the remaining XNUMX shaded triangles are placed in two rectangles, then it is clear that the resulting void, on the one hand, is equal to C in the square, and on the other, A + B, i.e. C \uXNUMXd A + B. The theorem has been proven. The mathematicians of ancient India noticed that to prove the Pythagorean theorem, it is enough to use the inside of the ancient Chinese drawing. In the treatise Sid-dhanta Shiromani (Crown of Knowledge), written on palm leaves, by the greatest Indian mathematician of the XNUMXth century, a drawing with the word “look!”, characteristic of Indian proofs, is placed in Bhaskara. Right-angled triangles are laid here with the hypotenuse outward and square C is shifted into the "bride's chair" square A plus square B. Particular cases of the Pythagorean theorem are found in the ancient Indian treatise "Sulva Sutra" (XNUMXth-XNUMXth centuries BC). Euclid's proof is given in sentence 1 of the book "Beginnings". Here, for proof, the corresponding squares are constructed on the hypotenuse and legs of a right triangle. "The Baghdad mathematician and astronomer of the 5th century, an-Nairizy (the Latinized name is Annaricius)," Voloshinov writes, "in the Arabic commentary on Euclid's "Principles" gave the following proof of the Pythagorean theorem. The square on the hypotenuse is divided by Annaricius into five parts, of which squares are composed on the legs.Of course, the equality of all the relevant parts requires proof, but we leave it to the reader for obviousness.It is curious that the proof of Annaricius is the simplest among the huge number of proofs of the Pythagorean theorem by the partition method: only 7 parts (or XNUMX triangles) appear in it. This is the smallest number of possible splits. Author: Samin D.K.
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