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.