Throughout the history of Mathematics, one of the most enduring and interesting challenges has been the calculation of the ratio between a circle’s circumference and diameter, which is popularly known by the Greek letter pi (π).
Probably we all know about the definition of pi, still I am quoting here from Wikipedia : “The number π is a mathematical constant, the ratio of a circle’s circumference to its diameter, approximately equal to 3.14159. It has been represented by the Greek letter ‘π’ since the mid18th century though it is also sometimes spelled out as ‘pi’ (π)”.
A little known verse of the Bible reads
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 about. (I Kings 7, 23; II Chronicles 4, 2.)
A ratio of 3:1 appears in the this biblical verse which is considered as one of the most ancient proof of calculation of π. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of ^{25}/_{8} = 3.125 and √10 = 3.162 have been traced to much earlier dates. Though in defense of Solomon’s craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. The Great Pyramid at Giza constructed in 2589–2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Due to this reason some historian thought that they had the knowledge of π. But controversially they found no other evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors and this connection to π was merely a coincidence.^{ }
The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of π including the ‘Biblical’ value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 × (^{8}/_{9})^{2} = 3.16 as a value for π. Various Babylonian and Egyptian writings suggest that each of the values 3, 3^{1}/_{6,} 3^{1}/_{7, }3^{1}/_{8 }were used (in different circumstances, of course)
The first theoretical calculation of a value of π was that of Archimedes of Syracuse (287212 BC). He worked out that 223/71 < π < 22/7. Archimedes’s results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed. Beginning with a hexagon, he worked all the way up to a polygon with 96 sides! Archimedes knew, what so many people to this day do not, that π does not equal ^{22}/_{7}, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.
Here is Archimedes’ argument.
Consider a circle of radius 1, in which we inscribe a regular polygon of 3 × 2^{n1} sides, with semiperimeter b_{n}, and superscribe a regular polygon of 3 × 2^{n1} sides, with semiperimeter a_{n}.
The diagram for the case n = 2 is on the right.
The effect of this procedure is to define an increasing sequence
b_{1} , b_{2} , b_{3} , …
and a decreasing sequence
a_{1} , a_{2} , a_{3} , …
such that both sequences have limit π.
Using trigonometrical notation, we see that the two semiperimeters are given by
a_{n} = K tan(π/K), b_{n} = K sin(π/K),
where K = 3 × 2^{n1}. Equally, we have
a_{n+1} = 2K tan(π/2K), b_{n+1} = 2K sin(π/2K),
and it is not a difficult exercise in trigonometry to show that
(1/a_{n} + 1/b_{n}) = 2/a_{n+1} . . . (1)
a_{n+1}b_{n} = (b_{n+1})^{2} . . . (2)
Archimedes, starting from a_{1} = 3 tan(π/3) = 3√3 and b_{1} = 3 sin(π/3) = 3√3/2, calculated a_{2} using (1), then b_{2} using (2), then a_{3} using (1), then b_{3} using (2), and so on until he had calculated a_{6} and b_{6}. His conclusion was that
b_{6} < π < a_{6} .
It is important to realize that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a_{6} and b_{6} from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.
Various mathematicians tried to produce accurate value of π by using polygon method. In ancient China, values for π included 3.1547 (around 1 AD), √10 (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556). Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygonbased iterative algorithm and used it with a 3,072sided polygon to obtain a value of π of 3.1416. Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96sided polygon. The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113, using Liu Hui’s algorithm applied to a 12,288sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920… remained the most accurate approximation of π available for the next 800 years.^{ }
The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD). Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value 3+√2/10 ≈ 3.14142.^{ }
The Persian astronomer Jamshīd alKāshī produced 16 digits in 1424 using a polygon with 3×2^{28} sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×2^{17} sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits. Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630, which remains the most accurate approximation manually achieved using polygonal algorithms.^{ }
The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for π. One of the earliest was that of Wallis (16161703)
2/π = (1.3.3.5.5.7. …)/(2.2.4.4.6.6. …)
and one of the bestknown is
π/_{4} = 1 – ^{1}/_{3} + ^{1}/_{5} – ^{1}/_{7} + ….
This formula is sometimes attributed to Leibniz (16461716) but is seems to have been first discovered by James Gregory (1638 1675).
These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while π arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.
From the point of view of the calculation of π, however, neither is of any use at all. In Gregory’s series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = ^{1}/_{20000}, and so we need about 10000 terms of the series. However, Gregory also showed the more general result
tan^{1} x = x – x^{3}/3 + x^{5}/5 – … (1 ≤ x ≤ 1) . . . (3)
from which the first series results if we put x = 1. So using the fact that
tan^{1}(^{1}/_{√3}) = π/_{6} we get
π/_{6} = (^{1}/_{√3})(1 – 1/(3.3) + 1/(5.3.3) – 1/(7.3.3.3) + …
which converges much more quickly. The 10^{th} term is 1/(19 × 3^{9}√3), which is less than 0.00005, and so we have at least 4 places correct after just 9 terms.
An even better idea is to take the formula
π/_{4} = tan^{1}(^{1}/_{2}) + tan^{1}(^{1}/_{3}) . . . (4)
and then calculate the two series obtained by putting first ^{1}/_{2} and the ^{1}/_{3} into (3).
Clearly we shall get very rapid convergence indeed if we can find a formula something like
π/_{4} = tan^{1}(^{1}/_{a}) + tan^{1}(^{1}/_{b})
with a and b large. In 1706 Machin found such a formula:
π/_{4} = 4 tan^{1}(^{1}/_{5}) – tan^{1}(^{1}/_{239}) . . . (5)
Machin reached 100 digits of π with this formula. Other mathematicians created variants, now known as Machinlike formulae, that were used to set several successive records for π digits. Machinlike formulae remained the bestknown method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device.^{ }
With a formula like this available the only difficulty in computing π is the sheer boredom of continuing the calculation. Needless to say, a few people were silly enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit. One of them, an Englishman named Shanks, used Machin’s formula to calculate π to 707 places, publishing the results of many years of labour in 1873. Shanks knew that π was irrational since this had been proved in 1761 by Lambert. Shortly after Shanks’ calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that ‘squaring the circle’ is impossible. The transcendentality of π implies that there is no ruler and compass construction to construct a square equal in area to a given circle.
Very soon after Shanks’ calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7’s. He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when D.F.Ferguson discovered that Shanks had made an error in the 528^{th} place, after which all his digits were wrong. In 1949 a computer was used to calculate π to 2000 places. In this and all subsequent computer expansions the number of 7’s does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.
We should say a little of how the notation π arose. Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference. David Gregory (1697) used π/r for the ratio of the circumference of a circle to its radius. The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states “3.14159 andc. = π”. Euler adopted the symbol in 1737 and it quickly became a standard notation.
We conclude with one further statistical curiosity about the calculation of π, namely Buffon’s needle experiment. If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/π. Various people have tried to calculate π by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got
π = ^{355}/_{113} = 3.1415929
which, incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously good, and the game is given away by the strange number 34080 of tosses. Kendall and Moran comment that a good value can be obtained by stopping the experiment at an optimal moment. If you set in advance how many throws there are to be then this is a very inaccurate way of computing π. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.
Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once. His estimate for π was thus given by
2 × 0.7857 / π = ^{1}/_{2}
from which he got the highly creditable value of π = 3.1428. He was not being serious!
It is almost unbelievable that a definition of π was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined π in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that π/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau’s dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau’s dismissal:
Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the unGerman style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.
G H Hardy replied immediately to Bieberbach in a published note about the consequences of this unGerman definition of π
There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one’s own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbach’s reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.
Not only in Germany did π present problems. In the USA the value of π gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.
Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)
The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!
In the twentieth century, computers took over the reigns of calculation, and this allowed mathematicians to exceed their previous records to get to previously incomprehensible results. In 1945, Ferguson discovered the error in William Shanks’ calculation from the 528th digit onward. Two years later, Ferguson presented his results after an entire year of calculations, which resulted in 808 digits of π. One and a half years later, Levi Smith and John Wrench hit the 1000digitmark . Finally, in 1949, another breakthrough emerged, but it was not mathematical in nature; it was the speed with which the calculations could be done. The ENIAC was finally completed and functional, and a group of mathematicians fed in punch cards and let the gigantic machine calculate 2037 digits in just seventy hours. Whereas it took Shanks several years to come up with his 707 digits, and Ferguson needed about one year to get 808 digits, the ENIAC computed over 2000 digits in less than three days!
John Wrench and Daniel Shanks found 100,000 digits in 1961, and the onemillionmark was surpassed in 1973. In 1976, Eugene Salamin discovered an algorithm that doubles the number of accurate digits with each iteration, as opposed to previous formulas that only added a handful of digits per calculation. Since the discovery of that algorithm, the digits of π have been rolling in with no end in sight. Over the past twenty years, six men in particular, including two sets of brothers, have led the race: Yoshiaki Tamura, Dr. Yasumasa Kanada, Jonathan and Peter Borwein, and David and Gregory Chudnovsky. Kanada and Tamura worked together on many π projects, and led the way throughout the 1980s, until the Chudnovskys broke the onebillionbarrier in August 1989. In 1997, Kanada and Takahashi calculated 51.5 billion digits in just over 29 hours, at an average rate of nearly 500,000 digits per second! The current record, set in 1999 by Kanada and Takahashi, is 68,719,470,000 digits. There is no knowing where or when the search for π will end. Certainly, the continued calculations are unnecessary. Just thirtynine decimal places would be enough to compute the circumference of a circle surrounding the known universe to within the radius of a hydrogen atom. Surely, there is no conceivable need for billions of digits. At the present time, the only tangible application for all those digits is to test computers and computer chips for bugs. But digits aren’t really what mathematicians are looking for anymore. As the Chudnovsky brothers once said: “We are looking for the appearance of some rules that will distinguish the digits of π from other numbers. If you see a Russian sentence that extends for a whole page, with hardly a comma, it is definitely Tolstoy. If someone gave you a million digits from somewhere in π, could you tell it was from π ? We don’t really look for patterns; we look for rules”. Unfortunately, the Chudnovskys have also said that no other calculated number comes closer to a random sequence of digits. Who knows what the future will hold for the almost magical number π ?
Hope you have enjoyed this long article. In next article a better chronology for π is coming up. Till then have a good time. And if you have anything else to share, don’t you forget to write up here.
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Arindam Bose
Scientific History Blog Writer • Art enthusiast and Illustrator • Amateur Photographer • Biker and Hiker • Beer Enthusiast • Electrical Engineer • Chicago

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