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Geometry History


A strange construction made by the ancient Persians to study the movement of the stars. An old compass. An ancient square and, under it, the figurative demonstration of Pythagorean theorem. A papyrus with geometric designs and the bust of the great Euclid. These are fundamental steps in the development of geometry. But long before the compilation of existing knowledge, men laid the foundations of geometry in the taste of experience. And they performed mental operations that would later be realized in the geometrical figures.

A measure for life

The Origins of Geometry (from Greek measure the earth) seem to match the needs of everyday life. Sharing fertile land on the banks of rivers, building houses, observing and predicting the movements of the stars are some of the many human activities that have always depended on geometric operations. Documents on the ancient Egyptian and Babylonian civilizations prove good knowledge of the subject, usually linked to astrology. In Greece, however, the genius of great mathematicians gave them a definite shape. From the Greeks before Euclid, Archimedes, and Apollonius, there is only the fragment of a Hippocratic work. And Proclo's summary commenting on Euclid's "Elements," a work dating from the fifth century BC, refers to Tales of Miletus as the introducer of Geometry in Greece by importation from Egypt.

Pythagoras named an important theorem about the triangle-rectangle, which inaugurated a new concept of mathematical demonstration. But while the 6th century BC Pythagorean school constituted a kind of philosophical sect that shrouded its knowledge in mystery, Euclid's "Elements" represent the introduction of a consistent method that has contributed more than twenty centuries to the progress of the sciences. It is the axiomatic system, which departs from the concepts and propositions admitted without demonstration (postulated the axioms) to logically construct everything else. Thus, three fundamental concepts — the point, the line, and the circle — and five related postulates serve as the basis for all so-called Euclidean geometry, useful even today, despite the existence of non-Euclidean geometries based on different (and contradictory) postulates. Euclid's.

The body as unity

The first units of measurement referred directly or indirectly to the human body: span, foot, step, arm, ulna. Around 3500 BC - when Mesopotamia and Egypt began building the first temples - their designers had to find more uniform and accurate units. They adopted the length of the body parts of a single man (usually the king) and with these measures built wooden and metal rulers, or knotted ropes, which were the first official length measurements.

Angles and figures

Among the Sumerians as well as the Egyptians, the primitive fields were rectangular in shape. The buildings also had regular floor plans, which forced architects to build many right angles (90 degrees). Though of reduced intellectual baggage, these men already solved the problem as a draftsman today. By means of two stakes driven into the earth they marked a straight line segment. Then they tied and stretched cords that worked in the manner of bars: two arcs of circumference cut and determine two points that, united, section perpendicular to the other straight, forming the right angles.

The most common problem for a builder is to draw, at a given point, the perpendicular to a line. The previous process does not solve this problem, where the right angle vertex is already determined beforehand. The old geometers solved it by means of three strings, placed to form the sides of a triangle-rectangle. These strings had lengths equivalent to 3, 4 and 5 units respectively. Pythagoras' theorem explains why: in every triangle-rectangle, the sum of the squares of the collars is equal to the square of the hypotenuse (opposite the right angle). E 32+42=52, ie 9 + 16 = 25.

Any trio of whole numbers or not that respect this relationship define triangles-rectangles, which in ancient times were standardized in the form of brackets.

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To measure surfaces

The priests charged with taxing the land probably began to calculate the length of the fields at a glance. One day, when observing workers paving a rectangular surface with square mosaics, some priest must have noticed that to know the total number of mosaics, it was enough to count those in a row and repeat that number as many rows as there were. Thus was born the formula of the rectangle area: multiply the base by the height.

Already to discover the area of ​​the triangle, the former inspectors followed an extremely geometric reasoning. To accompany it, just take a square or a rectangle and divide it into equal squares. Suppose the square has 9 "squares" and rectangle 12. These numbers then express the area of ​​these figures. Cutting the square into two equal parts, along the diagonal line, gives rise to two equal triangles whose area, of course, is half the area of ​​the square.

When faced with an uneven surface of the earth (neither square nor triangular), early mapmakers and surveyors appealed to the artifice known as triangulationStarting at any angle, they drew lines at all other visible angles of the field, so that it was completely divided into triangular portions, whose combined areas gave the total area. This method - still in use today - produced small errors when the terrain was not flat or had curved edges.

In fact, many lands follow the outline of a hill or the course of a river. And some buildings require a curved wall. Thus, a new problem presents itself: how to determine the length of a circle and the area of ​​a circle. By circumference is meant the periphery line of the circle, which is a surface. The old geometers observed that to draw circles, large or small, it was necessary to use a rope, long or short, and to rotate it around a fixed point, which was the stake embedded in the ground as the center of the figure. The length of this rope - known today as lightning It had something to do with the length of the circumference. By removing the rope from the stake and placing it over the circumference to see how many times it fit, they could see that it fit a little more than six and a quarter times. Whatever the length of the rope, the result was the same. Thus they drew some conclusions: a) the length of a circle is always about 6.28 times its radius; b) to know the length of a circumference, just check the length of the radius and multiply it by 6.28.

And the area of ​​the circle? The history of geometry explains it in a simple and interesting way. About 2000 BC, an Egyptian scribe named Ahmes wondered at the design of a circle in which he had traced the radius. Its purpose was to find the area of ​​the figure.

Tradition has it that Ahmes easily solved the problem: first, he thought of determining the area of ​​a square and calculating how many times that area would fit in the area of ​​the circle. Which square to choose? Any one? It seemed reasonable to take whatever side the figure's own radius was on. He did so, and proved that the square was contained in the circle more than 3 times and less than 4, or approximately, three times and a seventh (now we say 3.14 times). He then concluded that to know the area of ​​a circle, simply calculate the area of ​​a square built on the radius and multiply its area by 3.14.

The number 3.14 is basic in geometry and mathematics. The Greeks made it a little less inaccurate: 3,1416. Today, the symbol ("pi") represents this irrational number, already determined to be several dozen decimal places. Its name is only about two hundred years old and was taken from the first syllable of the word periphery, meaning circumference.

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New figures

Around 500 BC, the first universities were founded in Greece. Thales and his disciple Pythagoras gathered all the knowledge of Egypt, Etruria, Babylon, and even India to develop and apply to mathematics, navigation, and religion. Curiosity grew and books on geometry were in high demand. A bar soon replaced the rope and stake to draw circles, and the new instrument was incorporated into the arsenal of geometers. Knowledge of the universe was increasing rapidly, and the Pythagorean school even claimed that the earth was spherical rather than flat. New geometric constructions were emerging, and their areas and perimeters were now easy to calculate.

One of these figures was called polygonfrom greek polygonwhich means "many angles". Today even ship and aircraft routes are traced through advanced geometry methods, incorporated into radar equipment and other devices. Not surprisingly "since the days of ancient Greece, geometry has always been an applied science, that is, employed to solve practical problems. Of the problems that the Greeks were able to solve, two deserve mention: the calculation of the distance of an object to an observer and the calculation of the height of a building.

In the first case, to calculate, for example, the distance of a boat to the coast, a curious device was used. Two observers stood so that one could see the boat at a 90-degree angle to the shoreline and the other at a 45-degree angle. This done, the ship and the two observers were exactly at the vertices of an isosceles triangle, because the two acute angles measured 45 degrees each, so the collars were equal. It was enough to measure the distance between the two observers to know the distance from the boat to the shore.

Calculating the height of a building, a monument, or a tree is also very simple: a stake is vertically nailed to the earth, and the moment is expected when the extent of its shadow equals its height. The triangle formed by the stake, its shadow, and the line joining the ends of both are isosceles. Just measure the shadow to know the height.

Source: Knowing Encyclopedic Dictionary - April Cultural

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