An introduction to the modern theory of equations, by Florian Cajori.

CYCLIC EQUATIONS

207

Ex. 4. Show the impossibility of constructing, with ruler and compasses, the side of a cube, the volume of which is twice the volume of a
given cube.
(To construct a cube whose volume shall be double that of a given cube
is the problem known as the " Duplication of the Cube." It was one of
three problems upon which Greek mathematicians expended much effort.
Myth ascribes to it the following origin: The Delians were suffering from
a pestilence and were ordered by the oracle to double a certain cubical
altar. Thoughtless workmen constructed a cube with edges twice as long.
But brainless work like that did not pacify the gods. The error being
discovered, Plato was consulted on this "Delian problem." Through
him it received the attention of mathematicians.)
Ex. 5. Show the impossibility of trisecting by the aid of ruler and
compasses any given angle.
To trisect a given angle is the second of the three famous problems
first studied by Greek mathematicians. The third was the "Quadrature
of the Circle."
Let x be a complex number OA' of unit length. Let
[AOB =, IAOA' = A'OA" =-A"OB = 0.
Then x = cos + isin,
0 03
2 = Cos - + i sin2 -3 3
and x3 cos 0 + i sin. I
According to our problem we are given I, where x3 = OB, and we are to
show the impossibility of constructing OA' by ruler and compasses.
We are going to prove that equation I, as a rule, is irreducible. It is
sometimes reducible. For instance, when 0 = 90~, equation I gives x3=i,
which can be factored into (x +i) (X2-ix- 1), which factors are functions
in 0(L i). In this case the construction can be effected.
When the right member of I is an arbitrary number, that is, when 0 is
an arbitrary angle, then I is irreducible, else at least one of its roots could
be represented as a function of cos 0 and sin 0. By De Moivre's Theorem
the roots of I are
l = cos 3 + i sin
X2 + =.s 0 + 2
3 3
x2 = cos q5 -- +i sin + 47
X3 = COS + I Sill d+4d
3 3