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Archimedes of Syracuse
Archimedes of Syracuse (287 BC – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer.
Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series), and gave a remarkably accurate approximation of pi. He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.
In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between *265⁄*153 (approximately 1.7320261) and 1351*⁄*780 (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."
Archimedes also stated that a sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder) were placed on the tomb of Archimedes at his request.
Aryabhatta (476–550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhattiya (499 CE, when he was 23 years old) and the Arya-siddhanta.
After Aryabhattiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra
In Ganitapada 6, Aryabhatta gives the area of a triangle as:
"for a triangle, the result of a perpendicular with the half-side is the...