History of Conic Sections

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Chapter 10 : Quadratic Relations and Conic Sections
History of Conic Sections

History of Conic Sections
Apollonius of Perga (about 262-200 B.C.) was the last of the great mathematicians of the golden age of Greek mathematics. Apollonius, known as "the great geometer," arrived at the properties of the conic sections purely by geometry. His descriptions were so complete that he would have had little to learn about conic sections from our modern analytical geometry except for the improved modern notation. He did not, however, describe the properties of conic sections algebraically as we do today. It would take almost 2000 years before mathematicians would make great advances in the understanding of conic sections by combining both geometric and algebraic techniques. Apollonius defined the conic sections as sections of a cone standing on a circular base. The cone did not have to be a right cone, but could be slanted, or oblique. Apollonius noticed that all sections cut through such a cone parallel to its base were circles. He then extended the properties that he observed from these circles to ellipses and the other conic sections. He even solved the difficult problem of finding the shortest and longest distances from a given point to a conic section. These distances lie on lines called normals, which cut the curve of a conic section at right angles. Trying to read Apollonius' work, however, is not easy even with the best modern translation. Not only was his work very general in that it could be applied to many different situations involving conic sections, but it was also very long and rigorous, following the exacting principles of Euclidean proof. This is why the works of mathematicians like Hypatia, written several centuries later, were so important. Readers of Apollonius' Conics are better able to understand this text with the aid of these other texts that make the principles easier to understand. It is fortunate that scientists like Johannes Kepler, who...
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