The earliest known unambiguous examples of written records from Egypt and Mesopotamia, dating from about 3100 BCE, show that the ancients already had mathematical principles useful for surveying land areas, constructing buildings, and measuring storage vessels. And had started devising techniques. As early as the sixth century BCE, the Greeks gathered and expanded this practical knowledge and generalized it to the abstract subject now known as geometry, from the Greek words for measurement, geo (“earth”) and metron ( “Measurement”) by combining of the earth.
In addition to describing some of the achievements of the ancient Greeks, particularly Euclid’s logical development of geometry in the Elements, this article examines the application of geometry to astronomy, cartography, and painting from classical Greece to medieval Islam and Renaissance Europe. Some apps have been reviewed. It concludes with a brief discussion of extensions to non-Euclidean and multidimensional geometries in modern times.
Although many ancients, known and unknown, contributed to the subject, none equaled the impact of Euclid and his Elements of Geometry, a book now 2,300 years old and as painful and boring as the Bible. Diligent study is the goal. However, little is known about Euclid about Moses. In fact, the only thing that is known with any degree of confidence is that Euclid taught in the Library of Alexandria during the reign of Ptolemy I (323-285/283 BC). Euclid wrote not only on geometry but also on astronomy and optics and perhaps mechanics and music. Only elements, which were extensively copied and translated, have survived.
Euclid’s Elements was written so completely and clearly that it literally superseded the work of his predecessors. What is known of Greek geometry before this comes mainly from bits and pieces quoted by Plato and Aristotle and later mathematicians and commentators. Other valuable items they have preserved include some of the conclusions and general views of Pythagoras (c. 580-c. 500 bce) and his followers. The Pythagoreans convinced themselves that all things are in number, or owe to their relations. This theory gave the highest importance to mathematics in the research and understanding of the world. Plato developed a similar theory, and Pythagoras or philosophers inspired by Plato often wrote enthusiastically about geometry as the key to explaining the universe. Ancient geometry thus acquired an association with the sublime to complement its earthly origins and its reputation as a model of exact reasoning.
Ancient Geometry: Practical and Empirical
The origin of geometry lies in the concerns of everyday life. The traditional account, preserved in Herodotus’ History (5th century BCE), credits the Egyptians with inventing surveys to re-establish property values after the annual flooding of the Nile. Similarly, the eagerness to know the volume of tangible data derived from the need to assess tribute, store oil and grain, and build dams and pyramids. Even the three primitive geometrical problems of antiquity—doubling a cube, trisecting an angle, and squaring a circle, which will be discussed later—probably stemmed from practical matters, religious rites, time-keeping. and from the construction, respectively, are born in Pre-Greek Societies of the Mediterranean. and the fundamental subject of later Greek geometry, the theory of conic sections, for its general importance, and perhaps its origin, for its application to theory and astronomy.
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