The History of Pi
Summary
This article discusses the history of p. This account is divided into four parts here to facilitate the discussion. Through this discussion, the readers should gain a better understanding of the mysterious value of p and witness the wisdom and persistence of man.
I. Introduction
If you ask what p is one is sure to get answers like “circumference equals p times diameter.”, p is approximately 22/7 or 3.14. Though a simple constant p may seem, many mathematicians were devoted to studying it. Some mathematicians even spent their whole lives investigating this constant. So why is this constant so special?
Pi, represented by the Greek letter p, is the ration between the circumference and diameter. It has a history of 4000 years. This article aims to discuss the process of the quest for the value of p. The writer hopes that through this article, the readers will learn to admire the perseverance and wisdom of mankind and get a fuller picture of the magic of p.
For the sake of discussion, we will divide the body of the article into four parts, each on a different period on the time line.
II. Four Periods in the History of p
1. The First Step
How was Pi discovered? By Whom? Where and when?
As early as in 2000BC, people in ancient Babylon, Egypt, China and Israel discovered a fact: the circumference of a circle is directly proportional to the diameter. We shall discuss the discoveries by the Babylonians and Egyptians:
Ancient Babylon
The discovery originate from the calculations of perimeters. As noted from a piece of clay discovered in 1936, in the Ancient Babylonian period, (approximately 1900 – 1600 BC), the Babylonians believed that the perimeter of a regular hexagon is 0;57,36 (in base 60. This equals 96/100 = 24/25) times the circumference of the circumscribed circle:
perimeter of a regular hexagon = 24/25 ´ circumference of the circumscribed circle = 24/25 ´ p ´diameter
From this, we get an approximation of Pi:
p (Babylon) = 25/8 = 3.125
Egypt
The Egyptian discovery was due to the calculations of areas. In the Rhind Papyrus, there was a problem concerning Pi: “What is the area of a circular piece of land of diameter 9…… Take 8/8 of the diameter as the side of a square. Then the area of the square is equal to that of the circle.” This means:
A = (8d/9)2
and the approximation was:
p (Egypt) = (16/9)2 = 3.16049…
Some more facts
In China (at around 1200 BC): in the work “Zhou Bi Suan Jing” there was a saying of “circumference is three times the diameter”. This implies that the Chinese thought that p = 3.
In the Bible (around 500 BC), there was also a note on the value of Pi: ” He made a piece of copper. It was circular, 5 feet high, 10 feet in diameter and 30 feet in circumference.” (This was to describe the measurements of the Solomon temple) Therefore, people at that time thought that p = 3.
In this period, people did calculations only for daily purposes and rarely investigated the value of Pi just for the sake of it. Hence, most of their discoveries originated from experience (i.e., measurements). They were interested in Pi simply because of its applications in architecture and engineering. Thus, only the value of Pi mattered to them.
It was until 4 BC that people turned to ask how to find the value of Pi.
An important discovery: the Method of Exhaustion
Ancient Greece
Antiphon (approx. 430 BC) and Bryson (408 – 355 BC) invented a method to evaluate the area of plane shapes – the Method of Exhaustion. They also tried to calculate the area of a circle using this method.
“Construct a regular hexagon, doubling its number of sides repeatedly, this regular polygon will eventually ‘becomes’ a circle.”
Besides, Bryson had an innovative idea to calculate the area of a circle: the area of a circle is between those of its inscribed and inscribing polygons. This was probably the first time for man to use bounds to approach a value.
Unfortunately, as there was no way to evaluate numbers with large number of digits, they were not able to apply the method of exhaustion to find the value of Pi. Nevertheless, their idea of using exhaustion to approach a circle inspired many other mathematicians to follow their footsteps.
2. Development
Then, a development period followed and mathematicians at that time used polygons to find the value of Pi.
Ancient Greece:
An influential mathematician: Archimedes
Archimedes of Syracuse (287 – 212 BC) was the first mathematician to use systematic ways to find bounds and approximations to Pi. He applied the method of exhaustion on the perimeters of polygons. In Archimedes’ “The Measurement of the Circle”, he proposed three theorems on circles, and the third one was:
“The ratio of the circumference of any circle to its diameter is smaller than , but greater than ”
That is,
3.14084… < p < 3.14285…
He used the inscribing and inscribed regular polygons to determine the bounds for the circumference: Construct the inscribing and inscribed regular hexagons (Fig. 1) and take the values of their perimeters as the bounds. Then, form regular 12-gons from the hexagons, and repeating this process, using ratios and the Pythagoras’ Theorem, he found the perimeters of the inscribing and inscribed regular 96-gons of the circle (as in Fig. 2a and 2b). Finally, the ration of the circumferences to the diameters of the two regular 96-gons are the bounds for Pi.
| Figure 1：Hexagons inscribing and inscribed by the circle |
| Figure 2a: inscribing polygon | Figure 2b: inscribed polygon |
| (Archimedes’ methods to construct a 2n-gon from a n-gon, use proportions and the Pythagoras’ Theorem) |
This method of approaching Pi with polygons and method of exhaustion was pioneered by Archimedes and he obtained an approximation of Pi up to a precision of two decimal places (250 BC):
p = 3.14…
China:
The studies in mathematics in China were equally exciting. In the western Han Dynasty, an astronomer Liu Xin (50 BC – 20 AD) was chosen to develop a standard metric system. From a cylindrical bronze container he made, he estimated that p = 3.15. Later, in the Eastern Han Dynasty, another astronomer Zhang Heng (78 – 139 AD) estimated that (= 3.1622…) from his observation of stars, assuming the ratios of areas of the unit circle and its inscribing square to be 5 : 8. A better approximation of p = 3.155… was later found by another mathematician Wang Fan (217 – 257).
A milestone
At around 263 AD, a mathematician Liu Hui became the first person in China to develop a sound and solid method to calculate Pi－”Principle of Exhaustion for Circles”.
He first drew a circle of radius 10 and then he used the Pythagoras’ Theorem to determine the area of the regular hexagon inscribed by the circle. This gave the lower bound for the area of the circle. Then he extended to give a rectangle (Fig. 3) and this gave the upper bound of the area of the area of the circle. Using this as a basic procedure, he repeatedly doubled the number of sides of the regular polygon. He evaluated the areas of regular 12-gon, 24-gon…and 192-gon. This meant
| Figure 3: Let A be the area of circle, An be that of the inscribed regular n-gon, we have A2n < A < An + 2 (A2n – An), where 2 (A2n – An) is n times the area of the rectangle |
i.e.,
3.141024 < p < 3.142704
He wrote in “Notes on the Nine Chapters on the Mathematical Art”:
“The smaller pieces we divided into, the smaller the error. Repeating the division until we can no longer divide the circle, then the polygon will coincide with the circle.”
This implies that he also had the idea of the method of exhaustion as Antiphon and Bryson. His principle of exhaustion for circles is also similar to Archimedes’, in the sense that both of them made use of the method of exhaustion (repeatedly doubling the number of sides) and both are the only mathematicians to calculate the upper and lower bounds of Pi in their countries. But Liu Hui only used the inscribed polygon and its associated shapes while Archimedes used both the inscribing and inscribed polygons. Yet, it is believed that Liu Hui invented the principle of exhaustion for circles independently because of the difficulty in exchanging information across long distances at that time. In the end, Liu Hui found the area regular 3072-gon and get
p = 3927/1250 = 3.1416
which approximated p to three decimal places.
Reaching a new height
200 years after the discovery by Liu Hui, Zhu Chongzi (429 – 500) had also made great mathematical achievements. In a historical record, there was a note:
“At the end of the Sung Dynasty, Zhu Chongzi used a circle of diameter of 1 yard to find out that the upper bound of the circumference of the circle is 3.1415927 yards and the lower bound is 3.1415926 yards.”
(This indicates that the Chinese people at that time had the idea of digits already). Zhu Chongzi might have used Liu Hui’s principle of exhaustion for circles and his perseverance (as there was only very simple calculation tools available) to calculate the area of 24576-gon to get
3.1415926 < p < 3.1415927
This is a precision of 7 decimal places. This record of precision was broken only after around 1000 years.
On the other hand, he used p = 22/7 (= 3.14…) as a quick approximation and p = 355/113 (= 3.1415929) as a better approximation.
In fact, Archimedes was the first person to propose 22/7 as an approximation to p. For 355/113, it was first proposed by the German mathematician Valentin Otho in 1573. Hence, it is evident that the mathematical achievement in ancient China was indeed outstanding.
Marching On
In the 1000 years after Zhu Chongzi, European and Indian mathematicians continued to find better approximations to Pi.
For example, in 530 AD, the Indian mathematician Aryabhata (476 – 550) found in 530 the approximation
p = 62832/20000 = 3.1416
(He might have studied some Geek and Romanian scientific discoveries)
And in Europe, Leonardo of Pisa, Fibonacci (1180 – 1240) independently (without considering Archimedes’ methods) approximated Pi with 864/275, i.e.,
p = 3.141818…
However, during the Mid-Centuries, there was not much achievement in the studies of Pi and the accuracy of these approximations are pale in comparison with the Chinese and Indian ones. Besides, most of the European mathematicians just used the old method of approaching circles with polygons.
Geared towards Pi
Mathematician who spent their lives studying Pi:
The Dutch mathematician Ludolph Van Ceulen (1540 – 1610) began his quest for the value of Pi when he was 30 years old. In 1596, using Archimedes’ method, his persistence led him to the perimeter of regular 60 ´ 233-gon and hence p up to 20 decimal places. During his last year of life (1610), he finally arrived at the perimeter of regular 262-gon and p up to 35 decimal places. This set a record on hand calculation. Hence, in some Dutch and German textbooks, Pi is also called Ludolph number.
The last mathematician to use polygons to estimate Pi:
In 1621, the Dutch mathematician Wildebrod Snell (1580? – 1626) invented a more effective method. This made him to be able to find out more accurate approximations to Pi without doubling the number of sides of polygons. He divided each part of a polygon into three and add two sides to each of the original sides so as to better enclose the arcs. Using the identities , he got
3.14022 < p < 3.14160
Afterwards, in 1630, another mathematician used Snell’s method to evaluate p up to 39 decimal places. And he was the last mathematician for the 2000 years to use polygons to estimate Pi.
3. A Change
In this period, there came some breakthroughs and new formulae for searching for Pi.
Viète’s breakthrough
The French mathematician Francois Viète (1540 – 1603), using methods similar to Archimedes’ and trigonometric formulae, succeeded in 1579 in finding the perimeters of two 3 ´ 217(i.e. 393216)-gons and that 3.1415926535 < p < 3.1415926537. This gave p up to 9 decimal places. But he had a more important achievement – a breakthrough in the estimation of Pi – he was the first person to use infinite products to represent Pi.
In 1593, from one of Viète’s works:
The special point about this product is that it converges to p very rapidly. Using the first 25 terms, one can get Pi up to 15 decimal places. Viète’s method was:
Divide the polygons into triangles. He discovered that when the number of sides of the polygon increases, the ratio of the perimeter of a regular N-gon to that of a regular 2N-gon equals cos q. Then applying the half angle formula repeatedly, he got . When n increases, converges to x. Finally, using the formula and substituting , he get
After Viète, John Wallis (1616 – 1703) used another infinite product to represent p. In 1655, he used
for p. (Nowadays, we should write this as ).
In 1658, William Viscount Brouncker (1620 – 1684) turned this product into the first continued fraction of p:
The first arctangent series for Pi
On 15 February 1671, the British mathematician James Gregory (1638 – 1675) discovered the first arctangent series of p:
This series converges to p very slowly (we have to evaluate the first 5000 terms to get a precision of 5 decimal places). But his idea inspired succeeding mathematicians to use infinite series to evaluate p and this led to important development.
(The expansion offers a formula to calculate p using just addition and subtraction of fractions. This follows from that when we put x = 1, we have . Yet, Gregory himself did not attempt this.)
In 1674, German mathematician Gottfried Wilhelm Leibniz (1646 – 1716) independently found the first arctangent series (Leibniz’s series) for p:
Sharp’s breakthrough
British mathematician Abraham Sharp (1651 – 1742) was the first person to use Gregory’s series to calculate p. In 1699, he substituted to get
Every two terms in this series give a further decimal place in the precision of the approximation. Using this, one could easily get p up to 72 decimal places. And this marked the first estimation of p without using Archimedes’ methods.
The use of the symbol “p”
p is the sixteenth Greeter alphabet. The first mathematician to use p is William Oughtred (1574 – 1660). He used
to represent Pi, simply because p and d are the first letter of circumference and diameter in Greek respectively. In 1706, William Jones (1675 – 1749) used p for Pi for the first time. Later, Euler (1707 – 1783) used p for Pi in his “Introduction to the Analysis of the Infinite” in 1748 and since then p has been widely used to denote the ratio between the circumference and diameter.
p up to 100 decimal places
In 1706, the British mathematician John Machin (1680-1752) pioneered to use two arctangent series in an expression for Pi: and evaluate p up to 100 decimal places.
The contribution of Euler
Swiss mathematician Euler (1707 – 1783) made important contributions in the studies of Pi. He discovered many arctangent equations to determine Pi and many series on p and p2 (in his “Introduction to the Analysis of the Infinite”). In 1755, he discovered an arctangent series which converges very rapidly:
This series enables us to evaluate p up to 20 decimal places in an hour’s time.
Besides, Euler’s method of proving the irrationality of e inspired the German mathematician Johann Heinrich Lambert (1728 – 1777) to convert the tangent function to a continued fraction in 1761:
and thereby proved that p is irrational.
And then in 1775, Euler discovered the Euler’s formula, eip + 1 = 0. This formula was used in 1887 by German mathematician Ferdinand Lindemann (1852 – 1939) to prove that p is transcendental.
The hand calculation record of the value of Pi
William Shanks (1812 – 1882) used in 1873 had 707 decimal places. But it was later found that the proposed value was inaccurate starting from the 527th decimal place.
The last man in record of using pen and paper to evaluate p should be Ferguson. He spent his whole life, with the help of a hand calculator, in finding Pi up to 808 decimal places in 1947.
4. And More
Continuing from the above periods, in this period the huge advances in technology and the invention of computers sped up the calculation of Pi.
The first breakthrough by computer
In 1949, George Reitwiesner, John von Neumann and N. C. Metropolis used the ENIAC (Electronic Numerical Integrator and Computer) made in 1948 and Machin’s arctangent series to find p to 2037 decimal places in 70 hours.
The number of decimal places of p rocketed
French mathematician Francois Genuys used the Parisian IBM 704 computer to find up to 707 decimal places of p in 40 seconds. And later in 1958 he used Machin’s series and got 10000 decimal places in 100 minutes. On 29 July 1961, another mathematician Shanks got 100265 decimal places with a IBM 7090 computer. Then in 1973, Guilloud and Bouyer employed the Parisian CDC 7600 computer to find p to more than one million decimal places within 23 hours……
Until now, mathematicians are still searching for more effective ways or formulae to evaluate Pi, aiming to find more decimal places and the patterns in the digits.
III. Conclusion
Man has been searching for p for 4000 years. From the discovery of the fact that circumference is directly proportional to diameter, using polygons to approach circles to find Pi, discovering series and formulae for evaluation…..to the use of computers and technology, we are getting more and more digits of p.
Actually, we only need p up to 10 decimal places in daily or engineering calculations. Nowadays, we calculate p only to test speed of new computers. Yet, why are people still searching for p? The writer thinks that this is due to the curiosity of mathematicians to the infinite digits of p. Our mathematicians are eager to find out how many digits of Pi man can get. Is there any pattern in the digits of p? In order to set new records and test the limit of man, people are willing to spend time and effort to find the infinite and magic value of Pi and appreciate the beauty of mathematics.
http://www.geocities.com/walter_hung/misc/pi.htm :
p =3.
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 …
Infinite many digits of p?!
IV. References
Major references:
Jorg Arndt, Christoph Haenel (1998, 2000). p-Unleashed, p.165-208: “The History of p”, p.223-238: ” p Formula Collection”. Springer-Verlag Berlin Heidelberg New York.
David Blatner (translated in 1999 by Pan Endian). The Joy of p) Published in 1997 by Walker and Company). Taipei: Shang Ye Zhou Kan Chu Ban She Gu Fen You Xian Gong Shi.
Lie Zhi Jia, Jian Peihua and Huang Jiaming (2000). Shu Xue De Gu Shi, p. 25-36: “p de gu shi” (Huang Jingrong). Taipei: Jiu Zhang Chu Ban She.
Minor references:
Li Yan & Du Shiran (Translated by John N. Crossley & Anthony W. C. Lun) (1987). Chinese Mathematics A concise history, p.65-68: “The contribution of Liu Hui”, p.80-84: “The outstanding mathematician Zu Chongzhi in the period of the North and South Dynasties”. Oxford University Press.
Jiu Zhang Chu Ban She Bian Ji Bu (1981) Shu Xue Jia Zhuan Qi, p. 50-74: “Ke Xue Shang Chang Yong de Chang Shu – Yuan Zhou Lu“. Taipei: Jiu Zhang Chu Ban She.
Petr Beckmann (1982).A history of p (Pi) (Fifth Edition), p.198-199: “Chronological Table”. USA: The Golem Press.
http://www.geocities.com/walter_hung/misc/pi.htm
http://www.math.tku.edu.tw/mathhall/mathinfo/lwymath/piTOP.htm
http://www.nmns.edu.tw/New/Multimedia/china/A-2-2_display.htm