Doubling the Cube/Fallacious Proof

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Fallacious Proofs of Doubling the Cube

Edw. J. Goodwin

Doubling the dimensions of the cube octuples its contents, and doubling its contents increases its dimensions twenty-five plus one per cent.
-- Edw. J. Goodwin

Historical Note

The exercise to construct a cube double the volume of a given cube, using a compass and straightedge construction, was an exercise that the ancient Greeks failed to succeed in.

This was one of three such problems: the other two being Squaring the Circle and Trisecting the Angle.

There are several techniques available that use other tools, but these were considered unacceptably vulgar to the followers of Plato.

According to one version of the legend, the island of Delos was being ravaged by a plague. The people consulted the Oracle at Delphi, who instructed them that a new altar to Apollo had to be built which was twice the size of the existing one, which was cubic in shape.

Another version of the legend suggests that it was not a plague, but civil or political unrest causing the trouble.

The word size was misunderstood, and the new altar had each of the linear dimensions doubled, whereas apparently they should have doubled its volume instead. Hence, unfortunately, the gods were not appeased, and the desired abatement of the plague did not happen.

To quote Plato:

It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another.