The History of Geometry Geometry, from the ancient Greek “geo” meaning Earth and “metron” meaning measurement, arose as the field of knowledge dealing with spatial relationships. Geometry was revolutionized by Euclid, who introduced mathematical rigor still in use today. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. The earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, and ancient Babylonia from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus.
Greek Geometry The early history of Greek geometry is unclear, because no original sources of information remain and all of our knowledge is from secondary sources written many years after the early period. For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of the "axiomatic method", still in use today. One of the greatest names, is Thales of Miletus, a mathematician living in the 6th century BCE. He is regarded as the father of geometry and began the process of using deduction from first principles. It is believed that he travelled to Egypt and Babylon, picking up geometric techniques from these cultures, and he certainly would have had access to their work. Euclid is also a very famous name in the history of Greek geometry. He gathered the work of all of the earlier mathematicians and created his landmark work, 'The Elements,' surely one of the most published books of all time. In this work, Euclid set out the approach for geometry and pure mathematics generally, proposing that all mathematical statements should be proved through reasoning and that no empirical measurements were needed.
Indian Geometry Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry, a method of linking geometry and numbers first developed by the Greeks. They used ideas like the sine, cosine and tangent function, which relate the angles of a triangle to the relative lengths of its side, to survey the land around them, navigate the seas and even chart the heavens. Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 CE, contains the roots of modern trigonometry, including the first real use of sines, cosines, inverse sines, tangents and secants. One of the early Indian mathematicians to study geometry is Brahmagupta. Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established √10 (3.162277) as a good practical approximation for π (3.141593), and gave a formula, now known as Brahmagupta's Formula, for the area of a cyclic quadrilateral, as well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta's Theorem.
Chinese Geometry The earliest evidence of a systematic organization of regular shapes reproduced on material objects found in China dates from the third millennium BC. Painted motifs displaying geometrical patterns such as symmetrical arrangements of triangles, or circles have been found on ancient pottery pieces unearthed at Banpo. These designs show an early interest in spatial ordering. Chinese geometry developed without concern for geometrical definitions. But during the Han dynasty, Chinese mathematical knowledge was for the first time separated from other domains and recorded in the Jiuzhang suanshu. The geometry of Juizhang suanshu is not precisely an autonomous body of knowledge and consist essentially problems pertaining to planimetry, stereometry, and right-angled triangles.
Hindu Geometry The Hindu geometry originated in a very remote age in connection with the structure of the alters for the Vedic sacrifices. In the course of time, Hindu geometry grew beyond its original sacrificial purpose and began to be cultivated as a science for its own sake. This happened in the Vedic age when different schools of geometry were founded. More notable ones among them were the schools of Baudhayana, Apastamba and Katyayana. The Hindu geometry started in a brilliant way not only going by much in advance of the ancient Egyptian or Chinese geometry but also by anticipating some of the notable discoveries of the posterior Greek geometry. Hindu geometry did not make much progress in the post Vedic period as it ought to have done.