The very first concepts in geometry were acquired by people in ancient times. There was a need to determine the area of land, the volume of various vessels and premises and other practical needs. The history of the development of geometry, as a science, begins in ancient Egypt about 4 thousand years ago. Then the knowledge of the Egyptians was borrowed by the ancient Greeks, who used them mainly in order to measure the area of land. It is from Ancient Greece that the history of the emergence of geometry as a science originates. The ancient Greek word "geometry" is translated as "land surveying".
Greek scientists based on the discovery of many geometric properties were able to create a harmonious system of knowledge of geometry. The basis of geometric science was laid on the simplest geometric properties taken from experience. The remaining provisions of science were derived from the simplest geometric properties using reasoning. This whole system was published in its final form in the "Beginnings" of Euclid about 300 BC, where he outlined not only theoretical geometry, but also the foundations of theoretical arithmetic. The history of the development of mathematics also begins with this source.
However, Euclid’s work does not say anything about measuring volume, nor about the surface of a ball, nor about the ratio of the length of a circle to its diameter (although there is a theorem on the area of a circle). The history of the development of geometry was continued in the middle of the III century BC thanks to the great Archimedes, who was able to calculate the number Pi, and also was able to determine how to calculate the surface of the ball. To solve the mentioned problems, Archimedes applied methods that later formed the basis of higher mathematics methods . With their help, he could already solve the difficult practical problems of geometry and mechanics, which were important for navigation and for construction. In particular, he found ways to determine the centers of gravity and volumes of many physical bodies and was able to study the equilibrium of bodies of various shapes when immersed in a liquid.
Ancient Greek scientists conducted research on the properties of various geometric lines, important for the theory of science and practical applications. Apollonius in the II century BC made many important discoveries in the theory of conic sections, which remained unsurpassed for the next eighteen centuries. Appolonius applied the coordinate method to study conical sections. This method was further developed only in the XVII century by scientists Fermat and Descartes. But they used this method only to study flat lines. And only in 1748 the Russian academician Euler was able to apply this method to study curved surfaces.
The system developed by Euclid was considered immutable for more than two thousand years. However, in the future, the history of the development of geometry received an unexpected turn, when in 1826 the brilliant Russian mathematician N.I. Lobachevsky was able to create a completely new geometric system. In fact, the main provisions of his system differ from the provisions of Euclidean geometry in only one paragraph, but it is from this point that the main features of the Lobachevsky system follow. This is the position that the sum of the angles of a triangle in Lobachevsky’s geometry is always less than 180 degrees. At first glance it may seem that this statement is incorrect, but with small sizes of triangles modern measuring instruments do not allow to correctly measure the sum of its angles.
The further history of the development of geometry proved the correctness of the ingenious ideas of Lobachevsky and showed that the Euclidean system is simply unable to solve many questions of astronomy and physics, where mathematicians deal with figures of almost infinite dimensions. It is with the works of Lobachevsky that the further development of geometry, and with it higher mathematics and astronomy, is already connected.