Approximations of π - Early History

Early History

Some Egyptologists have concluded that the ancient Egyptians used an approximation of pi in their monuments, as the Great Pyramid of Giza was built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base (it is 1760 cubits around and 280 cubits in height). Others have argued that the ancient Egyptians had no concept of pi and would not have thought to encode it in their monuments. They argue, based on documents such as the Rhind papyrus, that the shapes of the pyramids are based on simple ratios of the sides of right angled triangles (the seked), however, the Rhind Papyrus in fact shows that the seked was derived from the base and height dimensions, and not the converse.

An Egyptian scribe named Ahmes wrote the oldest known text to imply an approximate value for pi. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period — though Ahmes stated that he copied a Middle Kingdom papyrus (i. e., from before 1650 BCE). In problem 48 the area of a circle was computed by approximating the circle by an octagon. The value of pi is never mentioned or computed, however. If the Egyptians knew of pi, then the corresponding approximation was 256/81.

As early as the 19th century BCE, Babylonian mathematicians were using π ≈ 25/8, which is about 0.5 percent below the exact value.

The Indian astronomer Yajnavalkya gave astronomical calculations in the Shatapatha Brahmana (c. 9th century BCE) that led to a fractional approximation of π ≈ 339/108 (which equals 3.13888..., which is correct to two decimal places when rounded, or 0.09 percent below the exact value).

In the third century BCE, Archimedes proved the sharp inequalities 223⁄₇₁ < π < 22⁄₇, by means of regular 96-gons; these values are 0.02 percent and 0.04 percent off, respectively. Later, in the second century CE, Ptolemy, used the value 377/120, which is correct to the equivalent of three decimal places.

The Chinese mathematician Liu Hui in 263 CE computed pi to between 3.141024 and 3.142708 by inscribing an 96-gon and 192-gon; the average of these two values is 3.141864, an error of less than 0.01 percent. However, he suggested that 3.14 was a good enough approximation for practical purposes. Later he obtained a more accurate result π ≈ 3927/1250 = 3.1416.