The Construction Problems of Antiquity

The classical construction problems of antiquity— squaring the circle, duplicating the cube, and trisecting the angle— cannot be solved by ruler and compass construction alone. As shown already by the Greeks, these constructions can be effected if one augments one's tool set. Hippias invented a curve called the quadratrix from a drawing of which the circle can be squared by further application of ruler and compass. Drawing the quadratrix, on the other hand, is another prolem. One can, however, duplicate the cube and trisect the angle using curves that can be drawn by mechanical devices. From the Greeks we have the conchoid of Nicomedes and the conic sections of Menæchmus; among the Arabs, al-Khayyami used conic sections to solve related algebraic problems; and in modern times we can point to Descartes who also relied on conics. The simplest device for angle trisection is a ruler on which one is allowed to make two marks to measure a fixed distance with.1 Today's history texts not only cite these results, but include all the details and there is no need to discuss them here. What the textbooks do not include and merits inclusion here is the proof of impossibility.


Minimal Polynomial Galois Theory Conic Section Irreducible Polynomial Degree Angle 
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© Springer Science+Business Media, LLC 2008

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