Doubling the cube
The doubling cube (cube volume doubling ) (also called Delos problem) is one of the classical problems of ancient mathematics.
According to legend, the inhabitants of the island of Delos interviewed during a plague epidemic 430 BC the oracle at Delphi for advice. There they were asked to double the cube-shaped altar in the temple of Apollo in volume. For ancient mathematicians, this meant that the side length of a cube with twice the volume using only ruler and compass should be constructed.
Only in the 19th century has been proved that this problem is unsolvable. The first proved Pierre Wantzel 1837. Todays evidence mostly use the Galois theory ( Évariste Galois, French mathematician ) and run in the core down to is that the irrational number not by integers, which can be expressed four basic arithmetic operations and square roots.
However, the doubling of the cube succeeded in ancient times using special curves (ie, not only with a compass and ruler). Hippocrates of Chios was the duplication of the cube reduced to a problem of the construction of relationships, and Archytas succeeded in their construction with a special curve. Also Eudoxus solved the problem ( its solution is lost ) and Menaechmus (as intersection of two conics based on Hippocrates transformation of the problem). Eratosthenes managed to Hippocrates based a geometric- mechanical solution he who chiseled in stone in the temple of Ptolemy in Alexandria.
The ancient sources are mainly to the Archimedes Comment by Eutocius, Plutarch and a fragment of Platonicus of Eratosthenes. Eratosthenes and Plutarch tracked the problem back to the oracle survey of the inhabitants of Delos. Plutarch adds that they had turned to Plato for advice, which they referred to Archytas, Eudoxus and Menaechmus. Their solution criticized Plato, according to Plutarch, since they make use of mechanical and non-geometrical methods, whereby he " geometrically " the exclusive use of compass and ruler said under.
Similar problems from the construction of altars (but with the problem of doubling a square instead of a cube ), there was in Vedic period in India and they gave to mathematical discussions occasion ( Sulbasutras ).