# An Introduction to Geometry

In geometry, a geometric object is a set of points, lines, or polygons, and the relations between them, in any notion of location or configuration, such as a plane figure, space, or a higher-dimensional space of an extended sort. Geometry has many subfields, including projective geometry, descriptive geometry, algebraic geometry, and combinatorial geometry. Geometry is one of the oldest fields of knowledge, and is included in the broad field of mathematics. The most basic questions in geometry are where, when, and why these geometries exist. Geometry can be divided into two main branches: Euclidean geometry and non-Euclidean geometry. Euclidean geometry is the branch of mathematics dealing with conic sections, and the geometry of the Euclidean plane and space, while non-Euclidean geometry is the branch of mathematics concerning the properties of these kinds of geometries. The field of geometry is one of the oldest fields of study. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia in the first millennium BC. The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, which date from the Third Dynasty of Egypt and the Second Dynasty of Ancient Egypt, respectively. Both texts on Egyptian geometry made use of geometry in their demonstration of many of their geometrical propositions. They were used by the Egyptians during the Middle Kingdom (2000–1800 BC) and later on in New Kingdom Egypt (1550–1070 BC) and were also used in the Second Intermediate Period (1070–945 BC). The ancient Greeks started to formalize geometry during the 5th century BC. They used geometry to calculate the lengths of the sides of a polygon, and to draw its interior. They used geometry to calculate the circumference of a circle, and to draw its circumference and its diameter. They used geometry to calculate the area and the volume of a figure, and to calculate the surface area and the volume of a solid. In ancient Greek philosophy, geometry was used to define nature and the physical universe, and also used to describe the human mind. Euclid's "Elements" helped to establish geometry as a separate branch of science. The Indian mathematician Mahavira (c. 6th century BC) was the first to state explicitly the principles of what is now called Euclidean geometry. He described the 5th century BC plane geometry of Eudoxus of Cnidus in a detailed commentary on the 13th century BC Indian philosophical work "Yukti-sastra" of the ancient Indian mathematicians Katyayana and Panini. The 9th century mathematician Brahmagupta used geometry to calculate the volume of a triangle and the area of a circle. He provided the first correct proofs for many geometric theorems. In the Islamic world, the first person to work on geometry fully was the Islamic polymath al-Haytham (Alhazen) (965–1040 AD). He was strongly influenced by Greek philosophy and in turn influenced the further development of geometry in Europe. In his book "Al-manasir wa al-hal", he discussed the general principles of proportion. In the book "Flatness of the Earth and its Rectification" he investigated the sphericity of the Earth and its possible rectification. In his book "The Configuration of the Universe" he proposed a model of the solar system, which was later used and developed by Ptolemy. The book also contains the first good description of the segment of a parabola and of a cycloid that can be found in any modern text book on geometry. Al-Haytham's book "Commentary on the Posterior" was a commentary on Ptolemy's "Geography" and was the first Arabic book to include a section on mathematics. In the 11th century, the Persian scholar Abu Nasr Mansur ibn al-Farkadain (also known as Albatenius) wrote a book called "Matheseos libri septem", which was the first book on mathematics written in Arabic. Abu Nasr's book included many mathematical results, including the first proof of the Pythagorean theorem. In the 13th century, the Italian mathematician Fibonacci (c. 1170–1250 AD) wrote his "Liber Abaci" ("Book of the Abacus"), a widely used mathematics book, which became a standard arithmetic text in Europe. In 1225, the Pisan mathematician Leonardo of Pisa (Leonardo Fibonacci) (Leonardo da Pisa) published his "Liber Quadratorum" ("Book of the Four Square") in Bologna. In 1238, the French philosopher and mathematician Nicole Oresme (c. 13