Using algebra in geometry

Algebra and geometry developed independently. Today, we could not possibly do without their intimate partnership. To deprive ourselves of the use of them together would render modern technology impossible at once. But the connection between algebra and geometry was not always obvious. Far from it, in fact. We now routinely use equations (algebra) to describe shapes (geometry). Everything from buildings to airplanes is constructed not just from blueprint drawings but from sets of equations describing the contours, surfaces, and structures. But the first people to do this sort of thing,,expressing geometrical shapes by equations, were the Chinese. A Chinese book of the third century AD called the Sea Island Mathematical Manual gives a series of geometrical propositions in algebraic form and describes geometrical figures by algebraic equations. Throughout Chinese history after that, if one wanted to consider geometry, algebra was regularly employed. These techniques spread westward to the Arabs when the famous mathematician al-Khwdrizmi was sent by the Caliph to be ambassador to Khazaria during the years 842 to 847. (Khazaria lay on the main trade routes between China and the West.) The first European who adopted the methods appears to have been Leonardo Fibonacci, who in his Practica Geometriae of 1220 used algebra in solving geometrical problems relating to the area of a triangle. Since the Chinese for so many centuries used algebra to study geometry, why did they not go on to invent analytic geometry, which is the great expression of those principles, in which every geometrical object and every geometrical operation can be referred to the realm of numbers? It was developed in Europe by the mathematicians Pierre Fermat and Rend Descartes in the seventeenth century. The reason seems to have been that the Chinese, strangely enough, never made the necessary study of conic sections, which gives such basic forms as, ellipses, parabolas, and hyperbolas. This was one of the Chinese blind spots. And furthermore, Needham says of the Europeans who did develop analytic geometry that they were 'reasoning from equations to geometrical figures; what the Chinese had always done was to transform geometrical figures into equations'. It is curious that the Chinese should have been a thousand years ahead with the bas' idea but that they should fail to push it home. It has often been said I'c that the key to modern science was the applying of mathematics to every aspect of the physical world - what is called 'the mathematization of nature'. Nowhere did the Chinese come so close as in using algebra to study geometry, and perhaps nowhere was their failure to follow through more crucial in dooming them never to achieve 'modern science'.

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