# Origins: 3.14159265…

#### Why did the ancients invent increasingly subtle and ingenious methods to arrive at an exact value of p? Human curiosity.

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How do you find the holy grail of mathematics?

You start with a circle, which is the easiest geometric shape to draw (just fix one end of a string in place and swing the other end around it, inscribing a circle). Then measure the circle’s perimeter (also known as the circumference) and the distance across its widest point (the diameter). Divide the circumference by the diameter—and you have that well-known but eternally daunting number,

That is part of the mystique of

Calculating the value of

The Temple craftsmen obviously obtained these numbers through direct measurement—perhaps using a rope—and they came up with a simple approximation of

According to a 4,000-year-old cuneiform tablet discovered in 1936, the Sumerians found the ratio of the perimeter of an inscribed hexagon to that of the circle to be 3456/3600, which factors out to 216/225. The Sumerians could thus 011measure any circle (by measuring an inscribed polygon and making the adjustment). Then they could measure the circle’s diameter—a simple straight line—and divide it into the circumference, producing an approximation of

In an ancient Egyptian mathematical treatise known as the Rhind Papyrus (c. 1650 B.C.E.), a scribe named Ahmes states that a certain circular field 9 units across (that is, with a diameter of 9) had an area of 64 units. Today, we know the relations between the diameter, circumference and area of a circle: Area equals ^{2}. Changing this equation around, we find that

Our next significant player is the Greek philosopher Antiphon. In the late fifth century B.C.E., he realized that if successive polygons were inscribed within a circle, doubling the number of sides each time, the difference between the polygon’s perimeter and the circle’s circumference would diminish toward zero (think of a circle as a polygon with an infinite number of sides). While Antiphon didn’t calculate

Two centuries later, Archimedes (c. 287–212 B.C.E.) inscribed a hexagon in a circle; then he doubled the sides until he had a 96-sided polygon inscribed in the circle. At the same time, he superscribed a similar series of polygons outside the circle. By this method, he found that

For almost 2,000 years, no one improved on Archimedes’s method of inscribed and superscribed polygons, though refinements were made in the calculation. The second-century C.E. Alexandrian astronomer Ptolemy, for instance, used Archimedes’s method to reach a value of 3.14167. And the method was invented independently by Indian and Chinese mathematicians. In the fifth century C.E., the Chinese mathematician Tsu Chung-Chih and his son Tsu Keng-Chih, using the polygon method, found that

The calculation of accurate trigonometric tables in the 16th century made the Archimedian approach much easier to pursue than before. The French lawyer and amateur mathematician François Viète (1540–1603) used trigonometry to calculate the perimeter of a polygon with 393,216 sides, pinpointing

But it was Isaac Newton’s development of calculus that reduced the calculation of

When it comes to endurance, nothing can beat a computer. In 1949, the primitive ENIAC computer, the first of the “giant brains,” was fed an algorithm for calculating

What’s the point of computing

How do you find the holy grail of mathematics? You start with a circle, which is the easiest geometric shape to draw (just fix one end of a string in place and swing the other end around it, inscribing a circle). Then measure the circle’s perimeter (also known as the circumference) and the distance across its widest point (the diameter). Divide the circumference by the diameter—and you have that well-known but eternally daunting number, p, or pi, which has a value of 3.14159265… That is part of the mystique of p: Whatever the size of the circle, the value […]

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