1768: Pi - The book of science

Pi
Proofs
Squaring the circle
Commentary

Johann Heinrich Lambert, Ferdinand von Lindemann mathematics

Pi

The history of π
(the ratio
of the circumference to the diameter of a circle)
is old.
The Egyptians knew of it;
the Babylonians knew of it;
the Greek mathematician Archimedes
used a polygon of ninety-six sides
to calculate it;
the Chinese mathematician Liu Hui knew of it
and used polygons of one hundred ninety-two sides
and three thousand seventy-two sides,
calculating their areas rather than perimeters.
Over the centuries, new methods for calculating π
have resulted in increasing accuracy—
but all decimal representations are approximations
of a number that is
never ending
and never repeating.

Proofs

In 1768, Johann Heinrich Lambert
proved that π is irrational—
it cannot be expressed as a ratio of whole numbers
and cannot have a repeating pattern.
In 1882, Ferdinand von Lindemann
proved that π is transcendental—
it cannot be a root of a polynomial equation
with rational coefficients.
One hopes, by now, inquiring minds have turned
to problems that can be solved.

Squaring the circle

The eye encompasses the obvious only
and often is mistaken.
Then the mind follows, convinced it should believe
what it has seen.
It doesn’t matter how hard we try
to turn a base metal into gold
to build a perpetual motion machine
or to square the circle,
human effort cannot controvert
the intricate and subtle
nature of our physical universe.
Seldom, if ever, do we exactly
hit the mark.

To “square the circle” is to find the length of the sides of a square with the same area as a circle of a given radius. The area of a circle of radius r is πr² , so success depends on the nature of π. The proofs of Lambert and Lindemann mean that we know that it is impossible for anyone to construct on a flat surface a square with the same area as a given circle using a compass and ruler.

The expression “squaring the circle” means trying to do the impossible.

Archimedes used the “method of exhaustion” to approximate the value of π. The work of both Archimedes and Liu Hui prefigured calculus; Archimedes wrote of infinitesimals and Liu Hui described the use of the limit.