Greek geometers devised ways to cleverly use the ruler and compass,
unaided by any other instrument, to perform all kinds of intricate and
beautiful constructions. They were so successful that it was hard to
believe they were unable to perform three little tasks which, at first
sight, appear to be very simple:
The first task demands that a cube be constructed having twice the
volume of a given cube. The second asks that any angle be divided into
three equal parts. The third requires the construction of a square
whose area is equal to that of a given circle. Remember, only a ruler
and compass are to be used!
Mathematicians, in Greek antiquity and throughout the Renaissance,
devoted a great deal of attention to these problems, and came up with
many brilliant ideas. But they never found ways of performing the above
three constructions. This is not surprising, for these constructions
are impossible! Of course, the Greeks had no way of knowing that fact,
for the mathematical machinery needed to prove that these constructions
are impossible—in fact, the very notion that one could prove a
construction to be impossible—was still two millennia away.
The final resolution of these problems, by proving that the requested
constructions are impossible, came from a most unlikely source: it was
a by-product of the arcane study of field extensions, in the upper
reaches of Modern Algebra.
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