Mathematicians generally believed that π was a transcendental but this ‘riddle
of the ages’was difficult to prove until Ferdinand von Lindemann used a
modification of a technique pioneered by Charles Hermite. Hermite had used it to
deal with the lesser problem of proving that the base of natural logarithms, e,
was transcendental (see page 26).
Following Lindemann’s result, we might think that the flow of papers from the
indomitable band of ‘circle-squarers’ would cease. Not a bit of it. Still dancing on
the sidelines of mathematics were those reluctant to accept the logic of the proof
and some who had never heard of it.
Constructing polygons
Euclid posed the problem of how to construct a regular polygon. This is a
symmetrical many-sided figure like a square or pentagon, in which sides are all
of equal length and where adjacent sides make equal angles with each other. In
his famous work the Elements (Book 4), Euclid showed how the polygons with 3,
4, 5 and 6 sides could be constructed using only our two basic tools.
Constructing an equilateral triangle
The polygon with 3 sides is what we normally call an equilateral triangle and
is particularly straightforward to construct. However long you want your triangle
to be, label one point A and another B with the desired distance in between.
Place the compass point at A and draw a portion of the circle of radius AB.