# Definition:Pi/Historical Note

## Historical Note on $\pi$ (Pi)

Every ancient society that considered circles was aware of $\pi$, although in general only as a rough approximation.

In the Old Testament, the implication is that $\pi = 3$:

*And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.*- -- $\text I$ Kings $7 : 23$

In ancient Babylon, the approximation $\dfrac {25} 8$ was used.

The Egyptian scribe Ahmes in the *Rhind Papyrus* used the approximation that the area of a circle equals the area of a square whose side is $\dfrac 8 9$ that of the diameter of the circle, leading to a value of $\pi$ of $\left({\dfrac {16} 9}\right)^2 = 3 \cdotp 16049 \ldots$

By calculating the areas of regular polygons of $96$ sides, Archimedes of Syracuse determined that $3 \dfrac {10} {71} < \pi < 3 \dfrac {10} {70}$, that is:

- $3 \cdotp 14085 \ldots < \pi < 3 \cdotp 142857 \ldots$

That last value:

- $3 \cdotp 142857 \ldots$

more often given as $\dfrac {22} 7$, is commonly used in schools as a good working approximation to $\pi$. This sequence is A068028 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

In binary notation it has the repeating pattern:

- $\pi \approx 11 \cdotp 00100 \, 1001 \ldots$

Archimedes also found more accurate approximations still.

Claudius Ptolemy used $\dfrac {377} {120}$, which is approximately $3 \cdotp 14166 \, 66 \ldots$

This sequence is A210622 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The **Zu Chongzhi fraction** is an exceptionally accurate approximation to $\pi$ (pi):

- $\pi \approx \dfrac {355} {113}$

whose decimal expansion is:

- $\pi \approx 3 \cdotp 14159 \, 292$

This sequence is A068079 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Zu Chongzhi and his son Zu Geng determined that:

- $3 \cdotp 14159 \, 26 < \pi < 3 \cdotp 14159 \, 27$

In the Indian tradition, $\sqrt {10} \approx 3.162$ was used.

Jamshīd al-Kāshī calculated $\pi$ to $16$ decimal places.

The **Ludolphine number** is the expression of the value of $\pi$ to $35$ decimal places:

- $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \ldots$

It was calculated by Ludolph van Ceulen between $1596$ and $1610$.

Improvements in trigonometric techniques allowed for better methods for calculating the digits of $\pi$.

Willebrord van Royen Snell calculated $34$ places using the same techniques that Ludolph van Ceulen used to calculate $14$.

Christiaan Huygens achieved $9$ places just by considering the geometry of the regular hexagon.

François Viète was the first to devise a formula for $\pi$, which he did in $1592$.

John Wallis was next, with Wallis's Product.

Isaac Newton devised a formula in $1666$, and Gottfried Wilhelm von Leibniz devised one in $1673$.

The latter is unfortunately too inefficient to be useful.

### Modern Developments

Since the middle of the $20$th century, considerable advances in the known digits of $\pi$ (pi) have been made using computers.

In $1945$ (or $1946$ -- sources are contradictory), D.F. Ferguson calculated $\pi$ to $610$ digits using a desk calculator, in the meantime discovering that the hitherto record-breaking $707$-digit work of William Shanks was incorrect from the $528$th place onwards.

During the course of $1947$, Ferguson extended his work to $710$ and $808$ digits.

In $1949$, working with John Wrench, this was once again extended to $1120$ digits.

By $1949$, electronic computers were being used.

Some of the record-breaking calculations are given in the following table:

Date | Contributor(s) | Computer | Time taken | Number of Digits |
---|---|---|---|---|

$1949$ | John Wrench, L.R. Smith and others | ENIAC | $70$ hours | $2037$ |

$1954$ | S.C. Nicholson and J. Jeenel | NORC | $13$ minutes | $3093$ |

$1958$ | George E. Felton | Ferranti Pegasus | $33$ hours | $10 \, 021$ |

$1961$ | Daniel Shanks and John Wrench | IBM 7090 | $8.7$ hours | $100 \, 265$ |

$1967$ | Jean Guilloud and M. Dichampt | CDC 6600 | $28$ hours | $500 \, 000$ |

$1983$ | Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura | HITAC M-280H | $33$ hours | $16 \, 777 \, 206$ (that is, $2^{24}$) |

$29$ April $2009$ | Daisuke Takahashi and others | T2K Open Supercomputer | $29 \cdotp 09$ hours | $2 \, 576 \, 980 \, 377 \, 524$ |

$11$ November $2016$ | Peter Trueb, using software by Alexander Yee | $4 \times$ Xeon E7-8890 v3 @ 2.50 GHz | $105$ days | $22 \, 459 \, 157 \, 718 \, 361$ |

### Indiana Pi Bill

The **Indiana Pi Bill** was an attempt by Edward J. Goodwin to legislate on the value of $\pi$.

Goodwin proposed bill #246 of the $1897$ sitting of the Indiana General Assembly, under the title:

*A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897.*

It somehow passed its first hearing.

However, before its second hearing, Clarence Abiathar Waldo, who happened to be present for the purpose of contributing towards another matter, was passed a copy of the bill and had the opportunity and capability to ensure that the bill was rejected.

### Notation

While the concept of $\pi$ (pi) dates back to antiquity, the standard symbol $\pi$ that is used for it dates back only to the $18$th century.

$\pi$ appears to have been first used for it for the first time by William Jones in $1706$, in his *Synopsis Palmariorum Matheseos*.

Jones followed earlier abbreviations for the Greek word **periphery** (**περιφέρεια**) by Oughtred and others.

Until fairly recently, it had been believed to date back only as far as Leonhard Paul Euler, who used it in his influential *Introductio in Analysin Infinitorum* of $1748$.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**pi** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**pi** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic of Shape: Archimedes