From Euclid to Spigotry
A Brief History of Geometry and p
Last modified: September 9, 2004
Thales (640-546 B.C.)
uses the abstract concepts of point and line.

Anaxagoras(500-428 B.C.) formulates the problem of "squaring the circle."

Hippias (5th century B.C.) invents the quadratrix and constructs, by a limiting process, two line segments whose lengths are in the ratio of p to 1.

Euclid (about 300 B.C.) writes the Elements

Postulates (as worded by H. S. M. Coxeter):
I. A straight line may be drawn from any point to any other point.
II. A finite straight line may be produced to any length in a straight line
III. A circle may be described with any centre at any distance from that centre.
IV. All right angles are equal.
V. If a straight line meet two other straight lines, so as to make the two interior angles on one side of it together less than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles.

Archimedes of Syracuse (287-212 B.C.) approximates the area of a circle using inscribed and circumscribed polygons and also devises a construction for trisecting an angle with the aid of a marked ruler. 

Pappus (3rd century A.D.) proves that Hippias' quadratrix solves the problem of "squaring the circle."

Tsu Chung-Chih (5th century A.D.) calculates pi to 8 decimal places of accuracy.

Vieta (1540-1603) develops a technique for approximating p
by repeatedly bisecting an angle and discovers how to reduce the problem of solving certain cubic equations to the problem of trisecting an angle.

Nilakantha and other Indian mathematicians(16th century) discover an infinite series that converges to p.

Descartes (1596-1650) invents analytic geometry.

Wallis, in 1655, discovers an infinite product for calculating p without extracting any square roots.

Newton, in 1665, invents the calculus and uses it to calculate pi to 16 decimal places.

Gregory, in 1671, rediscovers Nilakantha's infinite series for p, which becomes known as "Gregory's series."

Machin, in 1706, invents a series that converges much faster than Gregory's series and uses it to calculate 100 digits of p.
This remains "state of the art" until the mid 20th century.

Euler, in 1748, publishes numerous infinite series that converge to p.

Lambert, in 1766, proves that p is irrational.

Euler, in 1775, suggests that p is not merely irrational but transcendental.

Gauss, in 1796, proves that a regular heptadecagon can be constructed with compass and straightedge.

1700s: Many mathematicians make futile attempts to prove Euclid's Postulate V from the other four, but only end up by discovering alternative versions of the postulate.

Early 1800s: Gauss, Bolyai, and Lobachevsky independently discover hyperbolic geometry, which satisfies Euclid's first four postulates but not the fifth.

Liouville, in 1840, proves the existence of transcendental numbers.

Cantor, in 1871, proves that almost all real numbers are transcendental.

Hermite, in 1873, proves that e, the base of natural logarithms, is transcendental.

Lindemann, in 1882, proves that p is transcendental.

Ramanujan, in 1913, publishes without proof infinite series for 1/p that confound the mathematical world for decades.

An IBM 704 computer, in 1959, calculates 17,000 decimal digits of p.

1985-1995 Ideas of Euclid, Euler, and Gauss that have lain dormant for centuries are used to create new techniques for calculating large numbers of digits of p.

Alperin, in 2000, publishes the mathematical theory of origami constructions.

Kanada, in 2002, calculates 1.24 trillion digits of p.