We encounter geometry every second without even noticing it. Dimensions and distances, shapes and trajectories are all geometry. The meaning of the number π is known even to those who were geeks at school from geometry, and those who, knowing this number, are not able to calculate the area of a circle. A lot of knowledge from the field of geometry may seem elementary - everyone knows that the shortest path through a rectangular section is on the diagonal. But in order to formulate this knowledge in the form of the Pythagorean theorem, it took humanity millennia. Geometry, like other sciences, has developed unevenly. The sharp surge in Ancient Greece was replaced by the stagnation of Ancient Rome, which was replaced by the Dark Ages. A new surge in the Middle Ages was replaced by a real explosion of the 19th and 20th centuries. From an applied science, geometry has turned into a field of high knowledge, and its development continues. And it all began with the calculation of taxes and pyramids ...
1. Most likely, the first geometrical knowledge was developed by the ancient Egyptians. They settled on the fertile soils flooded by the Nile. Taxes were paid from the available land, and for this you need to calculate its area. The area of a square and a rectangle has learned to count empirically, based on similar smaller figures. And the circle was taken for a square, the sides of which are 8/9 of the diameter. At the same time, the number of π was approximately 3.16 - quite a decent accuracy.
2. The Egyptians engaged in the geometry of construction were called harpedonapts (from the word “rope”). They could not work on their own - they needed help-slaves, since to mark the surfaces it was necessary to stretch ropes of different lengths.
The pyramid builders did not know their height
3. The Babylonians were the first to use the mathematical apparatus for solving geometric problems. They already knew the theorem, which would later be called the Pythagorean Theorem. The Babylonians wrote down all tasks in words, which made them very cumbersome (after all, even the “+” sign appeared only at the end of the 15th century). And yet Babylonian geometry worked.
4. Thales of Miletsky systematized the then meager geometric knowledge. The Egyptians built the pyramids, but did not know their height, and Thales was able to measure it. Even before Euclid, he proved the first geometric theorems. But, perhaps, the main contribution of Thales to geometry was communication with the young Pythagoras. This man, already in old age, repeated the song about his meeting with Thales and its significance for Pythagoras. And another student of Thales named Anaximander drew the first map of the world.
Thales of Miletus
5. When Pythagoras proved his theorem, building a right-angled triangle with squares on its sides, his shock and shock of the students were so great that the students decided that the world was already known, it only remained to explain it with numbers. Pythagoras did not go far - he created many numerological theories that have nothing to do with either science or real life.
Pythagoras
6. Having tried to solve the problem of finding the length of the diagonal of a square with side 1, Pythagoras and his students realized that it would not be possible to express this length in a finite number. However, the authority of Pythagoras was so strong that he forbade the students to divulge this fact. Hippasus did not obey the teacher and was killed by one of the other followers of Pythagoras.
7. The most important contribution to geometry was made by Euclid. He was the first to introduce simple, clear and unambiguous terms. Euclid also defined the unshakable postulates of geometry (we call them axioms) and began to logically deduce all the other provisions of science, based on these postulates. Euclid's book "Beginnings" (although strictly speaking, it is not a book, but a collection of papyri) is the Bible of modern geometry. In total, Euclid proved 465 theorems.
8. Using Euclid's theorems, Eratosthenes, who worked in Alexandria, was the first to calculate the circumference of the Earth. Based on the difference in the height of the shadow cast by a stick at noon in Alexandria and Siena (not Italian, but Egyptian, now the city of Aswan), a pedestrian measurement of the distance between these cities. Eratosthenes received a result that is only 4% different from current measurements.
9. Archimedes, to whom Alexandria was no stranger, even though he was born in Syracuse, invented many mechanical devices, but considered his main achievement to be the calculation of the volumes of a cone and a sphere inscribed in a cylinder. The volume of the cone is one third of the volume of the cylinder, and the volume of the ball is two thirds.
Death of Archimedes. "Move away, you are covering the Sun for me ..."
10. Oddly enough, but for the millennium of Roman domination geometry, with all the flourishing of the arts and sciences in ancient Rome, not a single new theorem was proved. Only Boethius went down in history, trying to compose something like a lightweight, and even pretty distorted, version of the "Elements" for schoolchildren.
11. The dark ages that followed the collapse of the Roman Empire also affected geometry. The thought seemed to freeze for hundreds of years. In the 13th century, Adelard of Bartheskiy first translated "Principles" into Latin, and a hundred years later Leonardo Fibonacci brought Arabic numerals to Europe.
Leonardo Fibonacci
12. The first to create descriptions of space in the language of numbers began in the 17th century Frenchman Rene Descartes. He also applied the coordinate system (Ptolemy knew it in the 2nd century) not only to maps, but to all figures on a plane and created equations describing simple figures. Descartes' discoveries in geometry allowed him to make a number of discoveries in physics. At the same time, fearing persecution by the church, the great mathematician until the age of 40 did not publish a single work. It turned out that he was doing the right thing - his work with a long title, which is most often called “Discourse on Method,” was criticized not only by churchmen, but also by fellow mathematicians. Time proved that Descartes was right, no matter how trite it sounds.
René Descartes was rightly afraid to publish his works
13. Karl Gauss became the father of non-Euclidean geometry. As a boy, he independently learned to read and write, and once struck his father by correcting his accounting calculations. In the early 19th century, he wrote a number of works on curved space, but did not publish them. Now scientists were afraid not of the fire of the Inquisition, but of philosophers. At that time, the world was thrilled with Kant's Critique of Pure Reason, in which the author urged scientists to abandon strict formulas and rely on intuition.
Karl Gauss
14. In the meantime, Janos Boyai and Nikolai Lobachevsky also developed in parallel fragments of the theory of non-Euclidean space. Boyai also sent his work to the table, only writing about the discovery to friends. Lobachevsky in 1830 published his work in the magazine "Kazansky Vestnik". Only in the 1860s did the followers have to restore the chronology of the works of the entire trinity. It was then that it became clear that Gauss, Boyai and Lobachevsky worked in parallel, no one stole anything from anyone (and Lobachevsky was at one time attributed this), and the first was still Gauss.
Nikolay Lobachevsky
15. From the point of view of everyday life, the abundance of geometries created after Gauss looks like a game of science. However, this is not the case. Non-Euclidean geometries help solve many problems in mathematics, physics and astronomy.
