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I am very interested in the history of $\pi$. I am first trying to find out who calculated it. Many sources have different answers, from the ancient Egyptians, to Archimedes, to the Babylonians. I still can't find an answer to who first discovered $\pi$, or found a way to calculate it to any degree of accuracy. So who, or which group of people, were the first one(s) to discover/calculate $\pi$?

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    $\begingroup$ It's pretty trivial to notice that pi is about 3 based on stuff like wrapping a rope around a tree. I doubt that that you are going to find a first person who estimated it. $\endgroup$ – Ben Crowell Jan 24 '15 at 19:30
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    $\begingroup$ To what degree of accuracy? If you want $\frac{22}{7}$, you'll have a very different answer than if you were to want $3.14159$. $\endgroup$ – HDE 226868 Jan 24 '15 at 19:42
  • $\begingroup$ To any degree of accuracy $\endgroup$ – Anthony Pham Jan 24 '15 at 21:07
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    $\begingroup$ Nobody. Decimal expansion of $\pi$ has infinitely many digits and no pattern is known for them. Are you asking who first came up with a limit formula that can calculate it with any accuracy in principle? $\endgroup$ – Conifold Jan 24 '15 at 21:54
  • $\begingroup$ @conifold: even better, it is known that there is no pattern $\endgroup$ – Peter M. - stands for Monica Mar 7 '18 at 3:33
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It depends on the meaning of "calculate", since $\pi$ is a transcendental number it can not be "calculated" in the usual meaning of the word.

The first analytic formula (in the form of an infinite series) that in principle can calculate $\pi$ to any required accuracy is probably due to medieval Indian mathematician Madhava, who was first to conceive of infinite series explicitly, or one of his successors. In Europe this series was rediscovered by Leibniz, and is usually called Leibniz series in Western literature.

A semi-geometric "calculation" procedure capable in principle of producing arbitrary accuracy is much older. It consists of approximating a circle by inscribed and circumscribed polygons, and can be traced to ancient Greek orator Antiphon the Sophist. This method was mathematically justified by Eudoxus of Cnidus using what is now called method of exhaustion, and his justification is presented in Book XII of Euclid's Elements. Archimedes perfected the method in On the Measurement of the Circle. He proved rigorously that the ratio of the circle to the square on its radius was the same as the ratio of the circumference to the diameter, so it could be computed both ways.

Approximating the circumference with polygon perimeters is much simpler than approximating the circle area with polygon areas as Antiphon suggested. Using $96$-gons, Archimedes obtained what is now presented as the double estimate $3\frac{10}{71}<\pi<3\frac17$, although to Archimedes $\pi$ was not a number, and the result was phrased geometrically in terms of ratios of magnitudes. In 17th century Huygens refined the method further, and obtained the most accurate estimates to date, but after that analytic methods became more effective.

However, ancient Greeks also had a different concept of calculation, a purely geometric one, that was dominant in their time. A geometric magnitude was considered "calculated" if one could give a geometric construction for it. In the case of $\pi$ this meant constructing a square with area equal to the area of a given circle (or, after Archimedes, a segment with the length equal to half the circle's circumference). With straightedge and compass this is impossible (but it was only shown by Lindeman in 19th century), but Greeks also entertained more broad notions of "construction".

In this broader sense, the circle was first squared by Dinostratus around 350 BC using a curve, now known as the quadratrix. The curve itself was invented a century earlier by Hippias, who generated it by combining uniform linear and circular motions, and used it for angle trisection. In modern terms, from a segment of length $1$ it produces ("calculates") a segment of length $2/\pi$, and from that it is a simple matter to get a segment of length $\pi$ with straightedge and compass. Later Archimedes "calculated" $\pi$ in the same spirit using another curve generated by uniform linear and circular motions, the Archimedean spiral.

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  • $\begingroup$ Great point about the ultimate incalculability of transcendental numbers! $\endgroup$ – DukeZhou Jul 12 '17 at 21:45
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    $\begingroup$ I would interpret it as the first algorithm what could - at least, theoretically - calculate $\pi$ with any precision. $\endgroup$ – peterh says reinstate Monica Sep 10 at 10:44
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What do you mean "correctly or not"?

Here is a brief history. The Bible has a sentence which can be interpreted as implying that $\pi=3$.

The EXISTENCE of $\pi$ (the ratio of circumference to diameter) was rigorously proved by Archimedes. He also calculated it approximately. For many centuries it was called the "Archimedes number".

As centuries passed, they calculated it with larger and larger precision. Today it is known to more than a billion decimal digits and more. In 1990-th there was an interesting article in New Yorker magazine about the brothers Chudnovskii who evaluated the first billion digits. (With a huge computer they specially assembled for this purpose).

$\pi$ is a transcendental number, so you cannot "evaluate" it exactly (there is no finite or periodic decimal (or other) expression). This was proved by Lindemann in 1882.

Remark. Let me also add that there was a crank in 19-th century who "proved" that $\pi=4$. I think this is recorded in the Guinness book of records as the LEAST precise value of $\pi$ ever proposed. He offered his work as a gift to the State of Indiana. Indiana parliament had to decide whether to accept the gift. They could not decide. The decision was postponed. It is still postponed.

This gives some evil (or not informed) people a reason to say that "Indiana parliament LEGISLATED that $\pi=4$". Don't believe these people.

EDIT: My mistake: certainly "existence of $\pi$" was known to Euclid. Conifold: thanks for the correction. To address the same remark of Conifold: Euclid and Archimedes (and possibly Eudoxus) certainly understood perfectly what is a "real number" though they used somewhat different terminology of "proportions". Theory of proportions of Euclid is equivalent to the theory of real numbers.

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Since the question specifically says "to any accuracy," I will assume you mean approximations as well.

In this case, the first recorded approximation of $\pi$ comes from the Babylonians, who not only had an awareness of it as being a specific constant, had approximated it's value to $3\frac{1}{8}$ or $3.125$. This is recorded in a tablet fond near Susa dating from 2000 BC.

Both the Egyptians and Archimedes had a value for it but these date from much later.

From the Rhind Papyrus dating from around 1650 BC we find an Egyptian's scribe method for obtaining the area of a circle which is equivalent to using the value $\pi = 3\frac{1}{6}$ or $3.166$ repeating. This is perhaps nearly as good as the Babylonian approximation, but the Egyptians did not apparently have an awareness of it as being a specific constant, this is merely the effective value of it arising from their method.

From Archimedes there is given an approximation of $\pi$ more accurate than either the Egyptian or Babylonian, but this dates from almost 2000 years after the recorded Babylonian value.

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  • $\begingroup$ You're a little late to the party... Conifold's answer is good enough $\endgroup$ – Anthony Pham Jan 29 '15 at 23:10
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    $\begingroup$ Well, it's your question, but you did specifically ask about Archimedes vs Babylon vs Egypt and Conifold left out the Babylonians even though they had an approximation nearly a Millenia before Archimedes or Antiphon or Eudoxus or any other Greeks. Indeed, the mathematical center of the world was at one point, Babylon. After that, it shifts slowly to Egypt, and after that, Greece and the Mediterranean. The top voted answer neglected two whole periods, I felt. $\endgroup$ – iPherian Jan 30 '15 at 9:58
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There is a terrific summary here which includes the degree of accuracy with chronology, if that is what you are interested in.

Pre computer calculations of π

Mathematician           Date     Places Comments    
1   Rhind papyrus       2000 BC  1      3.16045 (= 4(8/9)2)
2   Archimedes          250 BC   3      3.1418 (average of the bounds)
3   Vitruvius           20 BC    1      3.125 (= 25/8)
4   Chang Hong          130      1      3.1622 (= √10)
5   Ptolemy             150      3      3.14166
6   Wang Fan            250      1      3.155555 (= 142/45)
7   Liu Hui             263      5      3.14159
8   Zu Chongzhi         480      7      3.141592920 (= 355/113)
9   Aryabhata           499      4      3.1416 (= 62832/20000)
10  Brahmagupta         640      1      3.1622 (= √10)
11  Al-Khwarizmi        800      4      3.1416
12  Fibonacci           1220     3      3.141818
13  Madhava             1400    11      3.14159265359
14  Al-Kashi            1430    14      3.14159265358979
15  Otho                1573     6      3.1415929
16  Viète               1593     9      3.1415926536
17  Romanus             1593    15      3.141592653589793
18  Ludolph Van Ceulen  1596    20      3.14159265358979323846
19  Ludolph Van Ceulen  1596    35      3.1415926535897932384626433832795029
20  Newton              1665    16      3.1415926535897932
21  Sharp               1699    71  
22  Seki Kowa           1700    10 
23  Kamata              1730    25 
24  Machin              1706    100 
25  De Lagny            1719    127     Only 112 correct
26  Takebe              1723    41  
27  Matsunaga           1739    50  
28  von Vega            1794    140     Only 136 correct
29  Rutherford          1824    208     Only 152 correct
30  Strassnitzky        1844    200 
31  Clausen             1847    248 
32  Lehmann             1853    261 
33  Rutherford          1853    440 
34  Shanks              1874    707     Only 527 correct
35  Ferguson            1946    620

Computer calculations of π

Mathematician        Date      Places   Type of computer
Ferguson            1947       710      Desk calculator
Ferguson, Wrench    1947       808      Desk calculator
Smith, Wrench       1949      1120      Desk calculator
Reitwiesner et al.  1949      2037      ENIAC
Nicholson, Jeenel   1954      3092      NORAC
Felton              1957      7480      PEGASUS
Genuys  Jan         1958     10000      IBM 704 
Felton  May         1958     10021      PEGASUS
Guilloud            1959     16167      IBM 704
Shanks, Wrench      1961    100265      IBM 7090
Guilloud, Filliatre 1966    250000      IBM 7030
Guilloud, Dichampt  1967    500000      CDC 6600
Guilloud, Bouyer    1973    1001250     CDC 7600
Miyoshi, Kanada     1981    2000036     FACOM M-200
Guilloud            1982    2000050
Tamura              1982    2097144     MELCOM 900II
Tamura, Kanada      1982    4194288     HITACHI M-280H
Tamura, Kanada      1982    8388576     HITACHI M-280H
Kanada, Yoshino, 
Tamura              1982    16777206    HITACHI M-280H
Ushiro, Kanada      1983    10013395    HITACHI S-810/20
Gosper              1985    17526200    SYMBOLICS 3670
Bailey  Jan         1986    29360111    CRAY-2
Kanada, Tamura      1986    33554414    HITACHI S-810/20
Kanada, Tamura      1986    67108839    HITACHI S-810/20
Kanada, Tamura, 
Kubo                1987    134217700   NEC SX-2
Kanada, Tamura      1988    201326551   HITACHI S-820/80
Chudnovskys         1989    480000000
Chudnovskys         1989    525229270
Kanada, Tamura      1989    536870898
Chudnovskys         1989    1011196691
Kanada, Tamura      1989    1073741799
Chudnovskys         1991    2260000000
Chudnovskys         1994    4044000000
Kanada, Tamura      1995    3221225466
Kanada              1995    4294967286
Kanada              1995    6442450938
Kanada, Takahashi   1997    51539600000     HITACHI SR2201 
Kanada, Takahashi   1999    206158430000    HITACHI SR
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