Piece of Pi History and Early Approximations

Did you know that the pi symbol (π) was only introduced in 1706, William Jones (1675-1749) used it in his Synopsis palmariorum matheseos, most likely after the initial letter of the Ancient Greek περιφέρεια ‎(periphéreia), meaning periphery (the line around the circle).

Before then, instead of π, the long Latin phrase “quantitas, in quam cum multiplicetur diameter, provenient circumferentia” had been used, meaning “the quantity which, when the diameter is multiplied by it, gives the circumference”. Clear, but quite a mouthful! In any case they had worked out that there was a mathematical constant there, which I reckon is already pretty cool.

Anyhow, π approximations! So who got close, and when? The answer may surprise you.

• The Babylonians  got to π = = 3.125 (sunbaked clay tablet found in 1936 at Susa).
• The Rhind Papyrus of Egypt (c. 1650 BCE) contains a solved problem that states that “the area of a circle of nine length units in diameter is the same as the area of a square whose side is eight units of length”, which (skipping some maths that is difficult to represent in a blogpost) comes to π = = 3.160 49
• The Greeks began with π = 3 for every day use (eek!); for more serious stuff they developed other values which were not much better, e.g. π = the square root of 10 = 3.162 2.
• In the 2nd century BCE, Hipparchus (c. 147 – after 127 BCE) did some extensive computations and proposed the value π = = 3.141 66…, which is not bad at all.
• Archimedes ( c. 287 – 212 BCE), regarded as the greatest scientist-mathematician of antiquity, worked out
  3.140 8… < π < 3.142 8…

Archimedes got to that value by applying his method for calculation of arc length to determine π. Beginning with regular hexagons – inscribed in, and circumscribed to a circle – and doubling the number of sides four times until he had a pair of regular 96-gons, he calculated the length of the perimeters of the successive polygons.

After that, no essentially new ideas for the calculation of π were suggested until the development of calculus towards the end of the 17th century! Of course, this had everything to do with the fall of the West Roman Empire in 476, which triggered the “Dark Ages” in Europe for a 1000 years. Mathematics and other sciences progressed very slowly in Europe during that time. But outside of Europe, things were actually moving along.

In India:

• Aryabatha, in 499, published π = 3.141 6…;
• Bhaskara (born in 1114), held that π = 3.141 56…;

unfortunately neither bothered telling how they got to those figures.

In China:

• Liu Hui in 263 CE published the limits
  3.140 24… < π < 3.142704… obtained for a pair of 96-gons (same method as Archimedes, but could he have been aware of that?); for a 3072-gon, he found π = 3.141 59… which is absolutely brilliant!
• Astronomer Tsu-Chung-chih (430-501) suggested π = = 3.141 592 9… which is correct to six decimal places, and was not to be bettered in Europe until the 16th century, more than a thousand years later.

In Persia:

• Jamshid Masud al-Kashi in 1424 published Risala al-muhitiyya (“Treatise on the Circumference”) with the results of his calculations on an inscribed substantial n-gon (3 x 3), arriving at π = 3.141 592 653 589 793 25… which is correct to sixteen decimal places, thereby surpassing all earlier determinations of π.

Back to Europe (after the end of the Dark Ages):

Ludolph van Ceulen (1540-1610), a fencing master teaching arithmetic, surveying and fortification at the engineering school at Leiden in Holland, got to 20 decimals of π in Van den Circkel (“About the Circle”, 1596), followed by 32 decimals in Arithmetische en Geometische fondamenten (“Arithmetic and Geometric fundamentals”), published posthumously in 161, and 35 decimals published in 1621 by his pupil Willebrord Snel,

π = 3.141 592 653 589 793 238 462 643 383 279 502 88

the last three digits of that were engraved as an epitaph on van Ceulen’s tombstone! Van Ceulen’s accomplishment so impressed his contemporaries that π was often called the Ludolphine constant.

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In conclusion… van Ceulen’s later accomplishments notwithstanding, it’s important to recognise the very significant achievements of the early Chinese and Persians over a thousand years earlier! (I’d include the Indians but they really should have shown their working)

Many awesome facts sourced from: Mathematics, from the Birth of Numbers (Jan Gulberg).