Why the circle ratio needed an approximation
Every circle has the same ratio between its circumference and its diameter. We write it as pi, roughly 3.14159. The symbol itself is modern: William Jones used the Greek letter in 1706, and Leonhard Euler’s adoption of it in the following decades made it standard. Long before that, builders, surveyors and scribes needed a working value, and a few mathematicians wanted to know how exactly it could be pinned down.
The number cannot be written as a fraction. That was proved only in the eighteenth century, by Johann Lambert, who showed what the ancient world suspected from the lack of any neat answer: pi belongs among the irrational numbers. So every early value below is an approximation, and the interesting part is how each culture reasoned its way to one.
Babylonian values
Many Babylonian calculations simply took the circumference to be three times the diameter, which is the rough rule of thumb most early builders used. A clay tablet from Susa, from the early second millennium BCE, is usually read as giving a ratio equivalent to 3 and 1/8, or 3.125. It appears in a comparison between a regular hexagon and the circle around it rather than in a direct measurement of a circle.
Babylonian arithmetic was written in the Mesopotamian sexagesimal system, base sixty, so the scribes would have expressed fractions differently from the way we do. The cuneiform place value notation made such fractions easy to handle, though the tablets do not announce that they are estimating a universal constant.
The Egyptian rule from the Rhind Papyrus
The Rhind Papyrus, a mathematical text copied around 1550 BCE from an older original, contains a rule for the area of a circle. Take the diameter, remove one ninth of it, and square the result. For a diameter of 9 units, that gives a square of side 8 and an area of 64.
Working back from that rule, the implied value of pi is 4 times (8/9) squared, which is 256/81, or about 3.1605. That is within about one percent of the true value, a respectable result for a problem solved with the fraction handling available in Egyptian hieroglyphic numerals and their unit-fraction system. The papyrus does not say how the rule was found, so modern explanations are reconstructions.
Archimedes and the polygon method
The first method that could, in principle, be pushed to any accuracy came from Archimedes in the third century BCE. In his short work on the measurement of a circle, he trapped the circumference between the perimeters of two polygons, one inside the circle and one outside it. Starting from a hexagon and doubling the number of sides repeatedly, he reached polygons of 96 sides.
His conclusion was that pi lies between 3 10/71 and 3 1/7. In decimals, 223/71 is about 3.1408 and 22/7 is about 3.1429, so the true value sits comfortably between them. The fraction 22/7 became a familiar everyday value, though it is a slight overestimate. What mattered historically was not the number but the idea of squeezing a value between an upper and lower bound.
Indian and Chinese refinements
In India, Aryabhata, writing around 500 CE, gave the ratio 62832 to 20000, which is 3.1416. In China, the mathematician Liu Hui, commenting in the third century CE on the classic text known as the Nine Chapters, used a polygon method similar in spirit to that of Archimedes. A 96-sided polygon gave him 3.14, and polygons with many more sides gave a value near 3.1416.
In the fifth century, Zu Chongzhi, working with his son, reportedly bounded pi between 3.1415926 and 3.1415927 and proposed the fraction 355/113 as a close rational value. That fraction equals about 3.14159292, correct to six decimal places, and it is remarkably compact. Zu’s own book is lost, and we know of the result mainly through later records, so details of his method are uncertain. Calculations of this kind in China were typically carried out with rod numerals on a counting surface, which made long chains of multiplication and square roots practical.
From polygons to series
Later work moved away from polygons. Around 1400 CE, Madhava of Sangamagrama and followers in southern India are credited with infinite series for pi, an approach that Gregory and Leibniz rediscovered in Europe in the seventeenth century. In the early fifteenth century, the Persian astronomer al-Kashi is generally credited with computing pi to roughly sixteen digits, and in 1610 Ludolph van Ceulen published a value with thirty-five digits, obtained from polygons with an enormous number of sides.
In 1882, Ferdinand von Lindemann proved that pi is transcendental, meaning it is not the root of any polynomial with whole-number coefficients. That result settled the old problem of squaring the circle with compass and straightedge as impossible.
What the sequence of values shows
| Culture | Approximate value | Decimal |
|---|---|---|
| Babylonian (Susa tablet) | 25/8 | 3.125 |
| Egyptian (Rhind Papyrus) | 256/81 | 3.1605 |
| Archimedes (upper bound) | 22/7 | 3.1429 |
| Archimedes (lower bound) | 223/71 | 3.1408 |
| Aryabhata | 62832/20000 | 3.1416 |
| Zu Chongzhi | 355/113 | 3.14159292 |
The table shows two different kinds of progress. The early values are practical rules that fit measurements well enough. From Archimedes onward, mathematicians were proving bounds and asking how far a method could be pushed. Each of these cultures reached its figure independently, with the tools and notation it had, and none of the values is a failed attempt at some other number; each was good enough for its purpose.