Birth, growth and computation of pi to ten trillion digits

The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has conjured up tremendous interest in mathematicians and non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed.

http://www.advancesindifferenceequations.com/content/2013/1/100 fractals, thermodynamics, mechanics, and electromagnetism. Pi hides in the rainbow, and sits in the pupil of the eye, and when a raindrop falls into water π emerges in the spreading rings. Pi can be found in waves and ripples and spectra of all kinds and, therefore, π occurs in colors and music. The double helix of DNA revolves around π . Pi has lately turned up in super-strings, the hypothetical loops of energy vibrating inside subatomic particles. Pi has been used as a symbol for mathematical societies and mathematics in general, and built into calculators and programming languages. Pi is represented in the mosaic outside the mathematics building at the Technische Universität Berlin. Pi is also engraved on a mosaic at Delft University. Even a movie has been named after it. Pi is the secret code in Alfred Hitchcock's 'Torn Curtain' and in 'The Net' starring Sandra Bullock. Pi day is celebrated on March  (which was chosen because it resembles .). The official celebration begins at : p.m., to make an appropriate . when combined with the date. In , the United States House of Representatives supported the designation of Pi Day. Albert Einstein was born on Pi Day ( March ).
Throughout the history of π , which according to Beckmann () 'is a quaint little mirror of the history of man' , and James Glaisher (-) 'has engaged the attention of many mathematicians and calculators from the time of Archimedes to the present day, and has been computed from so many different formula, that a complete account of its calculation would almost amount to a history of mathematics' , one of the enduring challenges for mathematicians has been to understand the nature of the number π (rational/irrational/transcendental), and to find its exact/approximate value. The quest, in fact, started during the pre-historic era and continues to the present day of supercomputers. The constant search by many including the greatest mathematical thinkers that the world produced, continues for new formulas/bounds based on geometry/algebra/analysis, relationship among them, relationship with other numbers such as π =  cos - (φ/), π / √ φ, where φ is the Golden section (ratio), and e iπ +  = , which is due to Euler and contains  of the most important mathematical constants, and their merit in terms of computation of digits of π . Right from the beginning until modern times, attempts were made to exactly fix the value of π , but always failed, although hundreds constructed circle squares and claimed the success. These amateur mathematicians have been called the sufferers of morbus cyclometricus, the circle-squaring disease. Stories of these contributors are amusing and at times almost unbelievable. Many came close, some went to tens, hundreds, thousands, millions, billions, and now up to ten trillion (  ) decimal places, but there is no exact solution. The American philosopher and psychologist William James (-) wrote in  'the thousandth decimal of Pi sleeps there though no one may ever try to compute it' . Thanks to the twentieth and twenty-first century, mathematicians and computer scientists, it sleeps no more. In , Hermann Schubert (-), a Hamburg mathematics professor, said 'there is no practical or scientific value in knowing more than the  decimal places used in the foregoing, already somewhat artificial, application' , and according to Arndt and Haenel (), just  decimal places would be enough to compute the circumference of a circle surrounding the known universe to within the radius of a hydrogen atom. Further, an expansion of π to only  decimal places would be sufficiently precise to inscribe a circle around the visible universe that does not deviate from perfect circularity by more than the distance across a single proton. The question has been repeatedly asked why so many digits? Perhaps the primary motivation for these computations is the human desire to break records; the extensive calculations involved http://www.advancesindifferenceequations.com/content/2013/ 1/100 have been used to test supercomputers and high-precision multiplication algorithms (a stress test for a computer, a kind of 'digital cardiogram'), the statistical distribution of the digits, which is expected to be uniform, that is, the frequency with which the digits ( to ) appear in the result will tend to the same limit (/) as the number of decimal places increases beyond all bounds, and in recent years these digits are being used in applied problems as a random sequence. It appears experts in the field of π are looking for surprises in the digits of π . In fact, 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 '? Some interesting observations are: The first  digits of π add up to  (which many scholars say is 'the mark of the Beast'); Since there are  degrees in a circle, some mathematicians were delighted to discover that the number  is at the th digit position of π . A mysterious  crop circle in Britain shows a coded image representing the first  digits of π . The Website 'The Pi-Search Page' finds a person's birthday and other well-known numbers in the digits of π . Several people have endeavored to memorize the value of π with increasing precision, leading to records of over , digits.
We believe that the study and discoveries of π will never end; there will be books, research articles, new record-setting calculations of the digits, clubs and computer programs dedicated to π . In what follows, we shall discuss the growth and the computation of π chronologically. For our ready reference, we also give some digits of π , π = .

.
About  BC. The meaning of the word sulv is to measure, and geometry in ancient India came to be known by the name sulba or sulva. Sulbasutras means 'rule of chords' , which is another name for geometry. The Sulbasutras are part of the larger corpus of texts called the Shrautasutras, considered to be appendices to the Vedas, which give rules for constructing altars. If the ritual sacrifice was to be successful, then the altar had to conform to very precise measurements, so mathematical accuracy was seen to be of the utmost importance. The sulbas contain a large number of geometric constructions for squares, rectangles, parallelograms and trapezia. Sulbas also contain remarkable approximations which gives √  = . . . . , and π ( - √ ) =   + √   , which gives π = . . . . . About  BC. Aryabhatta was born in  BC in Patliputra in Magadha, modern Patna in Bihar (India). He was teaching astronomy and mathematics when he was  years of age in  BC. His astronomical knowledge was so advanced that he could claim that the Earth rotated on its own axis, the Earth moves round the Sun and the Moon rotates http://www.advancesindifferenceequations.com/content/2013/1/100 round the Earth; incredibly he believed that the orbits of the planets are ellipses. He talks about the position of the planets in relation to its movement around the Sun. He refers to the light of the planets and the Moon as reflection from the Sun. He explains the eclipse of the Moon and the Sun, day and night, the length of the year exactly as  days. He calculated the circumference of the Earth as , miles, which is close to modern day calculation of , miles. In his Aryabhattiyam, which consists of the  verses and  introductory verses, and is divided into four padas or chapters (written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system), Aryabhatta included  verses giving  mathematical rules ganita on pure mathematics. He described various original ways to perform different mathematical operations, including square and cube roots and solving quadratic equations. He provided elegant results for the summation of series of squares and cubes. He made use of decimals, the zero (sunya) and the place value system. To find an approximate value of π , Aryabhatta gives the following prescription: Add  to , multiply by  and add to ,. This is 'approximately' the circumference of a circle whose diameter is ,. This means π = ,/, = .. It is important to note that Aryabhatta used the word asanna (approaching), to mean that not only is this an approximation of π , but that the value is incommensurable or irrational, i.e., it cannot be expressed as a ratio of two integers.
About  BC. Great pyramid at Gizeh was built around  BC in Egypt. It is one of the most massive buildings ever erected. It has at least twice the volume and thirty times the mass (the resistance an object offers to a change in its speed or direction of motion) of the Empire Sate Building in New York, and built from individual stones weighing up to  tons each. From the dimensions of the Great Pyramid, it is possible to derive the value of π , namely, π = half the perimeter of the base of the pyramid, divided by its height =  + / . . . . . About  BC. In a tablet found in  in Susa (Iraq), Babylonians used the value  π =   +  ()  , which yields π =  / = .. They were also satisfied with π = . About  BC. Ahmes (around - BC) (more accurately Ahmose) was an Egyptian scribe. A surviving work of Ahmes is part of the Rhind Mathematical Papyrus,  BC (named after the Scottish Egyptologist Alexander Henry Rhind who went to Thebes for health reasons, became interested in excavating and purchased the papyrus in Egypt in ) located in the British Museum since . When new, this papyrus was about  feet long and  inches high. Ahmes states that he copied the papyrus from a nowlost Middle Kingdom original, dating around  BC. This curious document entitled directions for knowing all dark things, deciphered by Eisenlohr in , is a collection of problems in geometry and arithmetic, algebra, weights and measures, business and recreational diversions. The  problems are presented with solutions, but often with no hint as to how the solution was obtained. In problem no. , Ahmes states that a circular field with a diameter of  units in area is the same as a square with sides of  units, i.e., π(/)  =   , and hence the Egyptian value of π is π =  ×    = . . . . , http://www.advancesindifferenceequations.com/content/2013/1/100 which is only very slightly worse than the Babylonians value, and in contrast to the latter, an overestimation. We have no idea how this very satisfactory result was obtained (probably empirically), although various justifications are available. Maya value of π was as good as that of the Egyptians. About  BC. The earliest Chinese mathematicians, from the time of Chou-Kong used the approximation π = . Some of those who used this approximation were mathematicians of considerable attainments in other respects. According to the Chinese mythology,  is used because it is the number of the Heavens and the circle.
About  BC. In the Old Testament (I Kings vii., and  Chronicles iv.), we find the following verse: ' Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about' . Hence the biblical value of π is / = . The Jewish Talmud, which is essentially a commentary on the Old Testament, was published about  AD. This shows that the Jews did not pay much attention to geometry. However, debates have raged on for centuries about this verse. According to some, it was just a simple approximation, while others say that '. . . the diameter perhaps was measured from outside, while the circumference was measured from inside' .
About  BC. Shatapatha Brahmana (Priest manual of  paths) is one of the prose texts describing the Vedic ritual. It survives in two recensions, Madhyandina and Kanva, with the former having the eponymous  brahmanas in  books, and the latter  brahmanas in  books. In these books, π is approximated by / = . . . . . About  BC. Anaxagoras of Clazomanae (- BC) came to Athens from near Smyrna, where he taught the results of the Ionian philosophy. He neglected his possessions in order to devote himself to science, and in reply to the question, what was the object of being born, he remarked: 'The investigation of the Sun, Moon and heaven' . He was the first to explain that the Moon shines due to reflected light from the Sun, which explains the Moon's phases. He also said that the Moon had mountains and he believed that it was inhabited. Anaxagoras gave some scientific accounts of eclipses, meteors, rainbows, and the Sun, which he asserted was larger than the Peloponnesus: this opinion, and various other physical phenomena, which he tried to explain which were supposed to have been direct action of the Gods, led him to a prosecution for impiety. While in prison he wrote a treatise on the quadrature of the circle. (The general problem of squaring a figure came to be known as the quadrature problem.) Since that time, hundreds of mathematicians tried to find a way to draw a square with equal area to a given circle; some maintained that they have found methods to solve the problem, while others argued that it is impossible. We will see that the problem was finally laid to rest in the nineteenth century.
About  BC. Hippocrates of Chios was born about  BC, and began life as a merchant. About  BC he came to Athens from Chios and opened a school of geometry, and began teaching, thus became one of the few individuals ever to enter the teaching profession for its financial rewards. He established the formula πr  for the area of a circle in terms of its radius. It means that a certain number π exists, and is the same for all circles, although his method does not give the actual numerical value of π . In trying to square the circle (unsuccessfully), Hippocrates discovered that two moon-shaped figures (lunes, bounded by pair of circular arcs) could be drawn whose areas were together equal to that of a right-angled triangle. Hippocrates gave the first example of constructing a rectilinear area equal to an area bounded by one or more curves. http://www.advancesindifferenceequations.com/content/2013/1/100 About  BC. Antiphon of Rhamnos (around - BC) was a sophist who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. Thus, he provided preliminary concept of infinitesimal calculus.
About  BC. Bryson of Heraclea was born around  BC. He was a student of Socrates. Bryson considered the circle squaring problem by comparing the circle to polygons inscribed within it. He wrongly assumed that the area of a circle was the arithmetical mean between circumscribed and inscribed polygons.
About  BC. Hippias of Elis was born about  BC. He was a Greek Sophist, a younger contemporary of Socrates. He is described as an expert arithmetician, but he is best known to us through his invention of a curve called the quadratrix (x = y cot(πy/)), by means of which an angle can be trisected, or indeed divided in any given ratio. It is not known whether Hippias realized that by means of his curve the circle could be squared; perhaps he realized but could not prove it. He lectured widely on mathematics and as well on poetry, grammar, history, politics, archeology and astronomy. Hippias was also a prolific writer, producing elegies, tragedies and technical treatises in prose. His work on Homer was considered excellent.
 BC. Aristophanes (- BC) in his play The Birds makes fun of circle squarers. Around  BC. Plato of Athens (around - BC) was one of the greatest Greek philosophers, mathematicians, mechanician, a pupil of Socrates for eight years, and teacher of Aristotle. He is famous for 'Plato's Academy' . 'Let no man ignorant of mathematics enter here' is supposed to have been inscribed over the doors of the Academy. He is supposedly obtained for his day a fairly accurate value for π = √  + √  = . . . . . About  BC. Eudoxus of Cnidus (around - BC) was the most celebrated mathematician. He developed the theory of proportion, partly to place the doctrine of incommensurables (irrationals) upon a thoroughly sound basis. Specially, he showed that the area of a circle is proportional to its diameter squared. Eudoxus established fully the method of exhaustions of Antiphon by considering both the inscribed and circumscribed polygons. He also considered certain curves other than the circle. He explained the apparent motions of the planets as seen from the earth. Eudoxus also wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the Sun, Moon and stars were attached, and which by their rotation produced the effects observed. In all, he required  spheres.
About  BC. Dinostratus (around - BC) was a Greek mathematician. He used Hippias quadratrix to square the circle. For this, he proved Dinostratus' theorem. Hippias quadratrix later became known as the Dinostratus quadratrix also. However, his demonstration was not accepted by the Greeks as it violated the foundational principles of their mathematics, namely, using only ruler and compass.
About  BC. Archimedes of Syracuse (- BC) ranks with Newton and Gauss as one of the three greatest mathematicians who ever lived, and he is certainly the greatest mathematician of antiquity. Galileo called him 'divine Archimedes, superhuman Archimedes'; Sir William Rowan Hamilton (-) remarked 'who would not rather have the fame of Archimedes than that of his conqueror Marcellus'?; Alfred North Whitehead (-) commented 'no Roman ever died in contemplation over a geometrical diagram'; Godfrey Harold Hardy (-) said ' Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not'; and http://www.advancesindifferenceequations.com/content/2013/1/100 Voltaire remarked 'there was more imagination in the head of Archimedes than in that of Homer' . His mathematical work is so modern in spirit and technique that it is barely distinguishable from that of a seventeenth-century mathematician. Among his mathematical achievements, Archimedes developed a general method of exhaustion for finding areas bounded by parabolas and spirals, and volumes of cylinders, parabolas, segments of spheres, and specially to approximate π , which he called as the parameter to diameter. His approach to approximate π is based on the following fact: the circumference of a circle lies between the perimeters of the inscribed and circumscribed regular polygons (equilateral and equiangular) of n sides, and as n increases, the deviation of the circumference from the two perimeters becomes smaller. Because of this fact, many mathematicians claim that it is more correct to say that a circle has an infinite number of corners than to view a circle as being cornerless. If a n and b n denote the perimeters of the inscribed and circumscribed regular polygons of n sides, and C the circumference of the circle, then it is clear that {a n } is an increasing sequence bounded above by C, and {b n } is a decreasing sequence bounded below by C. Both of these sequences converge to the same limit C. To simplify matters, suppose we choose a circle with the diameter , then from Figure  it immediately follows that a n = n sin π n and b n = n tan π n .
(  ) It is clear that lim n→∞ a n = π = lim n→∞ b n . Further, b n is the harmonic mean of a n and b n , and a n is the geometric mean of a n and b n , i.e., b n = a n b n a n + b n and a n = a n b n . It is interesting to note that during Archimedes time algebraic and trigonometric notations, and our present decimal system were not available, and hence he had to derive recurrence relations () geometrically, and certainly for him the computation of a  and b  must have been a formidable task. The approximation / is often called the Archimedean value of π , and it is good for most purposes. If we take the average of the bounds given in (), we obtain π = . . . . . The above method of computing π by using regular inscribed and circumscribed polygons is known as the classical method of computing π . It follows that an inscribed regular polygon of  n sides takes up more than  -/ n- of the area of a circle. Heron of Alexandria (about  AD) in his Metrica, which had been lost for centuries until a fragment was discovered in , followed by a complete copy in , refers to an Archimedes work, where he gives the bounds . . . . = , , < π < , , = . . . . .
Clearly, in the above right inequality, there is a mistake as it is worse than the upper bound / found by Archimedes earlier. Heron adds 'Since these numbers are inconvenient for measurements, they are reduced to the ratio of the smaller numbers, namely, /' . Archimedes' polygonal method remained unsurpassed for  centuries. Archimedes also showed that a curve discovered by Conon of Samos (around - BC) could, like Hippias' quadratrix, be used to square the circle. The curve is today called the Archimedean Spiral. About  BC. Daivajna Varahamihira (working  BC) was an astronomer, mathematician and astrologer. His picture may be found in the Indian Parliament along with Aryabhata. He was one of the nine jewels (Navaratnas) of the court of legendary king Vikramaditya I (- BC). In  BC, Varahamihira wrote Pancha-Siddhanta (The Five Astronomical Canons), in which he codified the five existing Siddhantas, namely, Paulisa Siddhanta, Romaka Siddhanta, Vasishtha Siddhanta, Surya Siddhanta and Paitamaha Siddhanta. He also made some important mathematical discoveries such as giving certain trigonometric formulae; developing new interpolation methods to produce sine tables; constructing a table for the binomial coefficients; and examining the pandiagonal magic square of order four. In his work, he approximated π as √ .  BC. Marcus Vitruvius Pollio (about - BC), a Roman writer, architect and engineer, in his multi-volume work De Architectura (On Architecture) used the value π =  / = ., which is the same as Babylonians had used , years earlier. He was the first to describe direct measurement of distances by the revolution of a wheel.
About  BC. Liu Xin (Liu Hsin) (about  BC- AD) was an astronomer, historian and editor during the Xin Dynasty (- AD). Liu created a new astronomical system, called Triple Concordance. He was the first to give a more accurate calculation of π as ., the exact method he used to reach this figure is unknown. This was first mentioned in the Sui shu (-). He also found the approximations ., . and ..
Around  AD. Liu Xin ( BC-AD ) was a Chinese astronomer, historian and editor during the Xin Dynasty (- AD). He was the son of Confucian scholar Liu Xiang (- BC). Liu created a catalog of , stars, where he used the scale of  magnitudes. He was the first in China to give a more accurate calculation of π as .. The method he used to reach this figure is unknown. http://www.advancesindifferenceequations.com/content/2013/1/100  AD. Brahmagupta (born  BC) wrote two treatises on mathematics and astronomy: the Brahmasphutasiddhanta (The Correctly Established Doctrine of Brahma) but often translated as (The Opening of the Universe), and the Khandakhadyaka (Edible Bite) which mostly expands the work of Aryabhata. As a mathematician he is considered as the father of arithmetic, algebra, and numerical analysis. Most importantly, in Brahmasphutasiddhanta he treated zero as a number in its own right, stated rules for arithmetic on negative numbers and zero, and attempted to define division by zero, particularly he wrongly believed that / was equal to . He used a geometric construction for squaring the circle, which amounts to π = √ . . Zhang Heng (- AD) was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman and literary scholar. He proposed a theory of the universe that compared it to an egg. 'The sky is like a hen's egg and is as round as a crossbow pellet. The Earth is like the yolk of the egg, lying alone at the center. The sky is large and the Earth is small' . According to him the universe originated from chaos. He said that the Sun, Moon and planets were on the inside of the sphere and moved at different rates. He demonstrated that the Moon did not have independent light, but that it merely reflected the light from the sun. He is most famous in the West for his rotating celestial globe, and inventing in  the first seismograph for measuring earthquakes. He proposed √  (about .) for π . He also compared the celestial circle to the width (i.e., diameter) of the earth in the proportion of  to , which gives π as ..
. Claudius Ptolemaeus (around - AD) known in English as Ptolemy, was a mathematician, geographer, astrologer, poet of a single epigram in the Greek Anthology, and most importantly astronomer. He made a map of the ancient world in which he employed a coordinate system very similar to the latitude and longitude of today. One of his most important achievements was his geometric calculations of semichords. Ptolemy in his famous Syntaxis mathematica (more popularly known by its Arabian title of the Almagest), the greatest ancient Greek work on astronomy, obtained, using chords of a circle and an inscribed -gon, an approximate value of π in sexagesimal notation, as    , which is the same as / = . . . . . Eutocius of Ascalon (about -) refers to a book Quick delivery by Apollonius of Perga (around - BC), who earned the title 'The Great Geometer' , in which Apollonius obtained an approximation for π , which was better than known to Archimedes, perhaps the same as /. . Wang Fan (-) was a mathematician and astronomer. He calculated the distance from the Sun to the Earth, but his geometric model was not correct. He has been credited with the rational approximation / for π , yielding π = ..
. Liu Hui (around -) wrote two works. The first one was an extremely important commentary on the Jiuzhang suanshu, more commonly called Nine Chapters on the Mathematical Art, which came into being in the Eastern Han Dynasty, and believed to have been originally written around  BC. (It should be noted that very little is known about the mathematics of ancient China. In  BC, the emperor Shi Huang of the Chin dynasty had all of the manuscript of the kingdom burned.) The other was a much shorter work called Haidao suanjing or Sea Island Mathematical Manual. In Jiuzhang suanshu, Liu Hui used a variation of the Archimedean inscribed regular polygon with  sides to approximate π as . and suggested / = . as a practical approximation.
About . Pappus of Alexandria (around -) was born in Alexandria, Egypt, and either he was a Greek or a Hellenized Egyptian. The written records suggest that, Pappus http://www.advancesindifferenceequations.com/content/2013/1/100 lived in Alexandria during the reign of Diocletian (-). His major work is Synagoge or the Mathematical Collection, which is a compendium of mathematics of which eight volumes have survived. Pappus' Book IV contains various theorems on circles, study of various curves, and an account of the three classical problems of antiquity (the squaring of the circle, the duplication of a cube, and the trisection of an angle). For squaring the circle, he used Dinostratus quadratrix and his proof is a reductio ad absurdum. Pappus is remembered for Pappus's centroid theorem, Pappus's chain, Pappus's harmonic theorem, Pappus's hexagon theorem, Pappus's trisection method, and for the focus and directrix of an ellipse.
. He Chengtian (-) gave the approximate value of π as ,/, = . . . . . . Tsu Ch'ung-chih (Zu Chongzhi) (-) created various formulas that have been used throughout history. With his son he used a variation of Archimedes method to find . < π < .. He also obtained a remarkable rational approximation /, which yields π correct to six decimal digits. In Chinese this fraction is known as Milü. To compute this accuracy for π , he must have taken an inscribed regular  ×  gon and performed lengthy calculations. Note that π = / can be obtained from the values of Ptolemy and Archimedes: He declared that / is an inaccurate value whereas / is the accurate value of π . We also note that π = / can be obtained from the values of Liu Hui and Archimedes. In fact, by using the method of averaging, we have . Bhaskara II or Bhaskaracharya (working ) wrote Siddhanta Siromani (crown of treatises), which consists of four parts, namely, Leelavati Bijaganitam, Grahaganitam and Goladhyaya. The first two exclusively deal with mathematics and the last two with astronomy. His popular text Leelavati was written in  AD in the name of his daughter. His contributions to mathematics include: a proof of the Pythagorean theorem, solutions of quadratic, cubic, and quartic indeterminate equations, solutions of indeterminate quadratic equations, integer solutions of linear and quadratic indeterminate equations, a cyclic Chakravala method for solving indeterminate equations, solutions of the Pell's equation and solutions of Diophantine equations of the second order. He solved quadratic equations with more than one unknown, and found negative and irrational solutions, provided preliminary concept of infinitesimal calculus, along with notable contributions toward integral calculus, conceived differential calculus, after discovering the derivative and differential coefficient, stated Rolle's theorem, calculated the derivatives of trigonometric functions and formulae and developed spherical trigonometry. He conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity. He speculated the nature of the number / by stating that it is 'like the Infinite, Invariable God who suffers no change when old worlds are destroyed or new ones created, when innumerable species of creatures are born or as many perish' . He gave several http://www.advancesindifferenceequations.com/content/2013/1/100 approximations for π . According to him ,/, is an accurate value, / is an inaccurate value, and √  is for ordinary work. The first value may have been taken from Aryabhatta. This approximation has also been credited to Liu Hui and Zu Chongzhi. He also gave the value / = ., which is of uncertain origin; however, it is the same as that by Ptolemy.
. Anicius Manlius Severinus Boethius (around -) introduced the public use of sun-dials, water-clocks, etc. His integrity and attempts to protect the provincials from the plunder of the public officials brought on him the hatred of the Court. King Theodoric sentenced him to death while absent from Rome, seized at Ticinum (now Pavia), and in the baptistery of the church there tortured by drawing a cord round his head till the eyes were forced out of the sockets, and finally beaten to death with clubs on October , . His Geometry consists of the enunciations (only) of the first book of Euclid, and of a few selected propositions in the third and fourth books, but with numerous practical applications to finding areas, etc. According to him, the circle had been squared in the period since Aristotle's time, but noted that the proof was too long.
. Abu Jafar Mohammed Ibn Musa al-Khwarizmi (around -) 'Mohammed the father of Jafar and the son of Musa' was a scholar in the academy Bait al-Hikma (House of Wisdom) founded by Caliph al-Mamun (-). His task (along with several other scholars) was to translate the Greek and Sanskrit scientific manuscripts. They also studied, and wrote on algebra, geometry and astronomy. There al-Khwarizmi encountered the Hindu place-value system based on the numerals , , , , , , , , , , including the first use of zero as a place holder in positional base notation, and he wrote a treatise around  AD, on what we call Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum (that is, al-Khwarizmi on the Hindu Art of Reckoning), a name given to the work by Baldassarre Boncompagni in , much changed from al-Khwarizmi's original text (of which even the title is unknown) is known. The French Minorite friar Alexander de Villa Dei, who taught in Paris around , mentions the name of an Indian king named Algor as the inventor of the new 'art' , which itself is called the algorismus. Thus, the word 'algorithm' was tortuously derived from al-Khwarizmi (Alchwarizmi, al-Karismi, Algoritmi, Algorismi, Algorithm), and has remained in use to this day in the sense of an arithmetic operation. This Latin translation was crucial in the introduction of Hindu-Arabic numerals to medieval Europe. Al-Khwarizmi used π = / in algebra, π = √  in geometry, and π = ,/, = . in astronomy. . Mahavira (-) in his work Ganita Sara Samgraha summarized and extended the works of Aryabhatta, Bhaskara, Brahmagupta and Bhaskaracharya. This treatise contains: a naming scheme for numbers from  up to   , formulas for obtaining cubes of sums; techniques for least common denominators (LCM), techniques for combinations n C r = n(n -)(n -) · · · (nr + )/r!, techniques for solving linear, quadratic as well higher order equations, arithmetic and geometric series, and techniques for calculating areas and volumes. He was the first person to mention that no real square roots of negative numbers can exist. According to Mahavira whatever is there in all the three worlds, which are possessed of moving and non-moving beings, all that indeed cannot exist without mathematics. He used the approximate value of π as √ . He also mentions that the approximate volume of a sphere with diameter d is (/)(d/)  , i.e., π = ., and exact volume is (/)(/)(d/)  , i.e., π = .. http://www.advancesindifferenceequations.com/content/2013/1/100 About . Franco von Lüttich (around -) claimed to have contributed the only important work in the Christian era on squaring the circle. His works are published in six books, but only preserved in fragments.
. Fibonacci (Leonardo of Pisa) (around -) after the Dark Ages is considered the first to revive mathematics in Europe. He wrote Liber Abbaci (Book of the Abacus) in . In this book, he quotes that 'The nine Indian numerals are. . . with these nine and with the sign  which in Arabic is sifr, any desired number can be written' . His Practica geometria, a collection of useful theorems from geometry and (what would eventually be named) trigonometry appeared in , which was followed five years later by Liber quadratorum, a work on indeterminate analysis. A problem in Liber Abbaci led to the introduction of the Fibonacci sequence for which he is best remembered today; however, this sequence earlier appeared in the works of Pingala (about  BC) and Virahanka (about  AD). In Practica geometriae, Fibonacci used a -sided polygon, to obtain the approximate value of π as / = . . . . .

.
Johannes Campanus (around -) was chaplain to three popes, Pope Urban IV, Pope Nicholas IV and Pope Boniface VIII. He was one of the four greatest contemporary mathematicians. Campanus wrote a Latin edition of Euclid's Elements in  books around . He used the value of π as /. About . Zhao Youqin (born ) used a regular polygon of  ×   sides to derive π = ..
About . Albert of Saxony (around -) was a German philosopher known for his contributions to logic and physics. He wrote a long treatise De quadratura circuli (Question on the Squaring of the Circle) consisting mostly philosophy. He said 'following the statement of many philosophers, the ratio of circumference to diameter is exactly /; of this, there is proof, but a very difficult one' .
. Madhava of Sangamagramma's (-) work has come to light only very recently. Although there is some evidence of mathematical activities in Kerala (India) prior to Madhava, e.g., the text Sadratnamala (about ), he is considered the founder of the Kerala school of astronomy and mathematics. Madhava was the first to have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series of sine, cosine, tangent and arctangent, which is This series is valid for - < x < , and also for x = . He also gave rational approximations of infinite series, tests of convergence of infinite series, estimate of an error term, early forms of differentiation and integration and the analysis of infinite continued fractions. He fully understood the limit nature of the infinite series. Madhava discovered the solutions of transcendental (transcends the power of algebra) equations by iteration, and found the approximation of transcendental numbers by continued fractions. He also gave many methods for calculating the circumference of a circle. The value of π correct to  decimal places is attributed to Madhava. However, the text Sadratnamala, usually considered as prior to Madhava, while some researchers have claimed that it was compiled by Madhava, gives the astonishingly accurate value of π correct to  decimal places. In , he calculated π to  decimal places, and later in  to  decimal places. For this, he used classical polygon method of  ×   sides.
. George Pürbach (-) whose real surname is unknown, was born in Pürbach, a town upon the confines of Bavaria and Austria. He studied under Nicholas de Cusa, and one of his most famous pupils is Regiomontanus. Pürbach wrote a work on planetary motions which was published in ; an arithmetic, published in ; and a table of eclipses, published in . He calculated tables of sines for every minute of arc for a radius of , units. This table was published in . He approximated π by the rational ,/,, which is exactly the same as given by Aryabhatta.
. Nicholas of Cusa (-) is often referred to as Nicolaus Cusanus and Nicholas of Kues (Cusa was a Latin place-name for a city on the Mosel). He was a German cardinal of the Roman Catholic Church, a philosopher, jurist, mathematician and an astronomer. Most of his mathematical ideas can be found in his essays, De Docta Ignorantia (Of Learned Ignorance), De Visione Dei (Vision of God) and On Conjectures. He made important contributions to the field of mathematics by developing the concepts of the infinitesimal and of relative motion. He gave the approximations of π as (/)( √  + √ ) and  √ / = . . . . . Nicholas thought this to be the exact value. Nicholas said, if we can approach the Divine only through symbols, then it is most suitable that we use mathematical symbols, for these have an indestructible certainty. He also said that no perfect circle can exist in the universe. In accordance with his wishes, his heart is within the chapel altar at the Cusanusstift in Kues.
. Johann Regiomontanus (Johannes Müller) (-) is considered as one of the most prominent mathematicians of his generation. He was the first to study Greek mathematical works in order to make himself acquainted with the methods of reasoning and results used there. He also well read the works of the Arab mathematicians. In most of this study, he compiled in his De Triangulis, which was completed in , however, was published only in . Regiomontanus used algebra to find solutions of geometrical problems. He criticized Nicholas of Cusa's approximations and methods to approximate the value of π and gave the approximation ..
About . Nilakanthan Somayaji's (around -) most notable work Tantrasangraha elaborates and extends the contributions of Madhava. He was also the author of Aryabhatiya-Bhashya, a commentary of the Aryabhatiya. Of great significance in Nilakanthan's work includes the inductive mathematical proofs, a derivation and proof of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, and in Sanskrit poetry the series which follows from Madhava's series () when x = . In the literature () is known as Gregory-Leibniz series. He also gave sophisticated explanations of the irrationality of http://www.advancesindifferenceequations.com/content/2013/1/100 π , the correct formulation for the equation of the center of the planets, and a heliocentric model of the solar system. If s n denotes the nth partial sum of (), then s  = , s  = . . . . , s  = . . . . , s , = . . . . , s , = . . . . and Roy North showed that s , = . (where underlined digits are incorrect) indicating an annoyingly slow convergence of the partial sums.
Since this is an alternating series, the error committed by stopping at the nth term does not exceed /(n + ) in absolute value. Thus, to compute π/ to eight decimals from () would require n >   terms. Hence, although it is only of theoretical interest, the expressions on the right are arithmetical, while π arises from geometry. We also note that the series () can be written as The following expansion of π is also due to Nilakanthan This series converges faster than (). Before . Leonardo da Vinci (-) was an Italian painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer. He briefly worked on squaring the circle, or approximating π .
. Michael Stifel (-) served in several different Churches at different positions; however, every time due to bad circumstances had to resign and flee. He made the error of predicting the end of the world on  October , and other time used a clever rearrangement of the letters LEO DECIMVS to 'prove' that Leo X was , the number of the beast given in the Book of Revelation. He was forced to take refuge in a prison after ruining the lives of many believing peasants who had abandoned work and property to accompany him to heaven. In the later part of his life, he lectured on mathematics and theology. He invented logarithms independently of Napier using a totally different approach. His most famous work is Arithmetica integra which was published in . This work contains binomial coefficients, multiplication by juxtaposition, the term 'exponent' , and the notation +, -and √ , and the opinion that the quadrature of π is impossible. According to him 'the quadrature of the circle is obtained when the diagonal of the square contains  parts of which the diameter of the circle contains ' . Thus, π  /. . Albrecht Dürer (-) was a famous artist and mathematician. His book Underweysung der Messung mit dem Zirckel und Richtscheyt provides measurement of lines, areas and solids by means of compass and ruler, particularly there is a discussion of squaring the circle.

.
Johannes Buteo (-), a French scholar published a book De quadratura circuli, which seems to be the first book that accounts the history of π and related problems. http://www.advancesindifferenceequations.com/content/2013/1/100 . Valentin Otho (around -) was a German mathematician and astronomer. In , he came to Wittenberg and proposed to Johannes Praetorius the Tsu Ch'ung-chih approximate value of π as /. . Tycho Brahe was an astronomer and an alchemist and was known for his most accurate astronomical and planetary observations of his time. His data was used by his assistant, Kepler, to derive the laws of planetary motion. He observed a new star in  and a comet in . In , when he was just , he lost his nose partially in a duel with another student in Wittenberg and wore throughout his life a metal insert over his nose. His approximation to π is / √  = . . . . . . Simon Duchesne finds π = (/)  = . . . . . About . Zhu Zaiyu (-), a noted musician, mathematician and astronomercalendarist, Prince of the Ming Dynasty, obtained the twelfth root of two. He also gave the approximate value of π as √ /. = . . . . . Around the same time Xing Yunlu adopted π as . and ., while Chen Jinmo and Fang Yizhi, respectively, took as . and /. . Simon van der Eycke (Netherland) published an incorrect proof of the quadrature of the circle. He approximated π as ,/ = . . . . . In , he gave the value ..
. Adriaen Anthoniszoon (-) was a mathematician and fortification engineer. He rediscovered the Tsu Ch'ung-chih approximation / to π . This was apparently lucky incident, since all he showed was that / > π > /. He then averaged the numerators and the denominators to obtain the 'exact' value of π .
. Francois Viéte (-) is frequently called by his semi-Latin name of Vieta. In relation to the three famous problems of antiquity, he showed that the trisection of an angle and the duplication of a cube problems depend upon the solution of cubic equations. He has been called the father of modern algebra and the foremost mathematician of the sixteenth century. In his  book, Supplementum geometriae, he showed . < π < ., i.e., gave the value of π correct to  places. For this, he used the classical polygon of  ×   = , sides. He also represented π as an infinite product For this, we note that which as k → ∞, and then x = π/ gives  π = cos π  cos π  cos π  cos π  · · · . http://www.advancesindifferenceequations.com/content/2013/1/100 Finally, note that The above formula () is one of the milestones in the history of π . The convergence of Vieta's formula was proved by Ferdinand Rudio (-) in . It is clear that Vieta's formula cannot be used for the numerical computation of π . In fact, the square roots are much too cumbersome, and the convergence is rather slow. It is clear that if we define a  = √ / and a n+ = √ ( + a n )/, then () is the same as a  a  a  · · · = /π . . Adrianus van Roomen (-), more commonly referred to as Adrianus Romanus, successively professor of medicine and mathematics in Louvain, professor of mathematics at Würzburg, and royal mathematician (astrologer) in Poland, proposed a challenge to all contemporary mathematicians, to solve a certain th degree equation. The Dutch ambassador presented van Roomen's book to King Henry IV with the comment that at present there is no mathematician in France capable of solving this equation. The King summoned and showed the equation to Vieta, who immediately found one solution to the equation, and then the next day presented  more. However, negative roots escaped him. In return, Vieta challenged van Roomen to solve the problem of Apollonius, to construct a circle tangent to three given circles, but he was unable to obtain a solution using Euclidean geometry. When van Roomen was shown proposer's elegant solution, he immediately traveled to France to meet Vieta, and a warm friendship developed. The same year Rooman used the classical method with   sides, to approximate π to  correct decimal places.
. Joseph Justus Scaliger (-) was a religious leader and scholar. He is known for ancient Greek, Roman, Persian, Babylonian, Jewish and Egyptian history. In his work, Cyclometrica elementa duo he claimed that π is equal to √ . . Ludolph van Ceulen (-) was a German who emigrated to the Netherlands. He taught Fencing and Mathematics in Delft until , when he moved to Leiden and opened a Fencing School. In , he was appointed to the Engineering School at Leiden, where he spent the remainder of his life teaching Mathematics, Surveying and Fortification. He wrote several books, including Van den Circkel (On The Circle, ), in which he published his geometric findings, and the approximate value of π correct to  decimal places. For this, he reports that he used classical method with  ×   , i.e., ,,, sides. This book ends with 'Whoever wants to, can come closer. ' . Ludolph van Ceulen (-) in his work De Arithmetische en Geometrische fondamenten, which was published posthumously by his wife in , computed π correct to  decimal places by using classical method with   sides. This computational feat was considered so extraordinary that his widow had all  digits of die Ludolphsche Zahl (the Ludolphine number) was engraved on his tombstone in St. Peter's churchyard in Leiden. The tombstone was later lost but was restored in . This was one of the last major attempts to evaluate π by the classical method; thereafter, the techniques of calculus were employed. http://www.advancesindifferenceequations.com/content/2013/1/100 . Willebrord Snell (Snellius) (-) was a Dutch astronomer and mathematician. At the age of , he is said to have been acquainted with the standard mathematical works, while at the age of , he succeeded his father as Professor of Mathematics at Leiden. His fame rests mainly on his discovery in  of the law of refraction, which played a significant role in the development of both calculus and the wave theory of light. However, it is now known that this law was first discovered by Ibn Sahl (-) in . Snell cleverly combined Archimedean method with trigonometry, and showed that for each pair of bounds on π given by the classical method, considerably closer bounds can be obtained. By his method, he was able to approximate π to seven places by using just  sides, and to van Ceulen's  decimal places by using polygons having only   sides. The classical method with such polygons yields only two and fifteen decimal places.
. Yoshida Mitsuyoshi (-) was working during Edo period. His  work named as Jinkoki deals with the subject of soroban arithmetic, including square and cube root operations. In this work, he used . for π .
. Christoph (Christophorus) Grienberger (-) was an Austrian Jesuit astronomer. The crater Gruemberger on the Moon is named after him. He used Snell's refinement to compute π to  decimal places. This was the last major attempt to compute π by the Archimedes method.
. William Oughtred (-), an English mathematician offered free mathematical tuition to pupils, which included even Wallis. His textbook, Clavis Mathematicae (The Key to Mathematics) on arithmetic published in  was used by Wallis and Newton amongst others. In this work, he introduced the × symbol for multiplication, and the proportion sign (double colon ::). He designated the ratio of the circumference of a circle to its diameter by π/δ. His notation was used by Isaac Barrow (-) a few years later, and David Gregory (-). Before him, mathematicians described π in round-about ways such as 'quantitas, in quam cum multipliectur diameter, proveniet circumferential' , which means 'the quantity which, when the diameter is multiplied by it, yields the circumference' .
. Grégoire de Saint-Vincent (-), a Jesuit, was a mathematician who discovered that the area under the hyperbola (xy = k) is the same over [a, b] as over [c, d] when a/b = c/d. This discovery played an important role in the development of the theory of logarithms and an eventual recognition of the natural logarithm. In , Nicolaus Mercator (Kauffmann) (-) wrote a treatise entitled Logarithmo-technica, and discovered the series however, the same series was independently discovered earlier by Saint-Vincent. In his book, Opus geometricum quadraturae circuli et sectionum coni he proposed at least four methods of squaring the circle, but none of them were implemented. The fallacy in his quadrature was pointed out by Huygens. . René Descartes (-) was a thoughtful child who asked so many questions that his father called him 'my little philosopher' . In , he published his Discourse http://www.advancesindifferenceequations.com/content/2013/1/100 on Method, which contained important mathematical work, and three essays, Meteors, Dioptrics and Geometry, produced an immense sensation and his name became known throughout Europe. The rectangular coordinate system is credited to Descartes. He is regarded as a genius of the first magnitude. He was one of the most important and influential thinkers in human history and is sometimes called the founder of modern philosophy. After his death, a novel geometric approach to approximate π was found in his papers. His method consisted of doubling the number of sides of regular polygons while keeping the perimeter constant. In modern terms, Descartes' method can be summarized as and hence The sequence {x k } generated by the above recurrence relation converges to /π . . John Wallis (-) in  was appointed as Savilian professor of geometry at the University of Oxford, which he continued for over  years until his death. He was the most influential English mathematician before Newton. In his most famous work, Arithmetica infinitorum, which he published in , he established the formula This formula is a great milestone in the history of π . Like Viéte's formula (), Wallis had found π in the form of an infinite product, but he was the first in history whose infinite sequence involved only rational operations. In his Opera Mathematica I (), Wallis introduced the term continued fraction. He rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity, specially showed that - > ∞. He had great ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed. On  December , he when in bed, occupied himself in finding the integral part of the square root of  ×   ; and several hours afterward wrote down the result from memory. Two months later, he was challenged to extract the square root of a number of  digits; this he performed mentally, and a month later he dictated the answer which he had not meantime committed to writing. Wallis' life was embittered by quarrels with his contemporaries including Huygens, Descartes, and the political philosopher Hobbes, which continued for over  years, ending only with Hobbes' death. Hobbes called Arithmetica infinitorum 'a scab of symbols' , and claimed to have squared the circle. It seems that to some, individual's quarrels give strength, encouragement and mental satisfaction. To derive (), we note that http://www.advancesindifferenceequations.com/content/2013/1/100 sin n x dx satisfies the recurrence relation Thus, in view of I  = π/ and I  = , we have From these relations, a termwise division leads to Now, it suffices to show that We know that for all x ∈ (, π/) the inequalities sin m- x > sin m x > sin m+ x hold. Thus, an integration from  to π/ gives I m- ≥ I m ≥ I m+ , and hence Further, from (), we have thus, it follows that Finally, a combination of () and () immediately gives (). If we define a n =  -/(n)  , then () is equivalent to a  a  a  · · · = /π . We also note that . William Brouncker, nd Viscount Brouncker (-) was one of the founders and the second President of the Royal Society. His mathematical contributions are: reproduction of Brahmagupta's solution of a certain indeterminate equation, calculations of the lengths of the parabola and cycloid, quadrature of the hyperbola which required approximation of the natural logarithm function by infinite series and the study of generalized http://www.advancesindifferenceequations.com/content/2013/1/100 continued fractions. He undertook some calculations to verify formula (), and showed that . . . . < π < . . . . , which is very satisfactory. He also converted Wallis' result () into the continued fraction () Neither of the expressions (), and (); however, later has served for an extensive calculation of π . Another continued fraction representation of π which follows from the series () is . Christiaan Huygens (-) is famous for his invention of the pendulum clock, which was a breakthrough in timekeeping. He formulated the second law of motion of Newton in a quadratic form, and derived the now well-known formula for the centripetal force, exerted by an object describing a circular motion. Huygens was the first to derive the formula for the period of an ideal mathematical pendulum (with massless rod or cord), T = π /g. For the computation of π , he gave the correct proof of Snell's refinement, and using an inscribed polygon of only  sides obtained the bounds . < π < ., for the same accuracy the classical method requires almost , sides.
. Muramatsu Shigekiyo (-) published Sanso, or Stack of Mathematics, in which he used classical polygon method of   sides to obtain π = ..
. Sir Isaac Newton (-), hailed as one of the greatest scientist-mathematicians of the English-speaking world, had the following more modest view of his own monumental achievements: '... to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me' . As he examined these shells, he discovered to his amazement more and more of the intricacies and beauties that lay in them, which otherwise would remain locked to the outside world. At the age of , he succeeded Barrow as Lucasian professor of mathematics at Cambridge. About him, Aldous Huxley (-) had said 'If we evolved a race of Isaac Newtons, that would not be progress. For the price Newton had to pay for being a supreme intellect was that he was incapable of friendship, love, fatherhood and many other desirable things. As a man he was a failure; as a monster he was superb' . Newton made some of the greatest discoveries the world ever knew at that time. Newton discovered: . The nature of colors. . The law of gravitation and the laws of mechanics. . The fluxional calculus. Most of the history books say that to compute π Newton used the series which for x = / gives however, he actually used twenty-two terms to obtain  decimal places of the following series Later, he wrote 'I am ashamed to tell you to how many figures I carried these computations, having no other business at the time' . His result was not published until  (posthumously). Using analysis and geometry, the series () can be obtained as follows: From Figure , the equation of the upper half circle is y = x / (x) / . Thus, binomial expansion gives Thus, the area of the sector ABD is (integrating the above series from  to /) () Also, from geometry the area of the sector ABD is Equating () and (), we immediately get (). . Thomas Hobbes of Malmesbury (-) was an English philosopher, best known today for his work on political philosophy. He also contributed in several other diverse fields, including history, geometry, the physics of gases, theology, ethics and general philosophy. He approximated π by  / = ., which was refuted by Huygens and Wallis. In , he also gave the approximation √ . . James Gregory (-) published two books Vera circuli et hyperbolae quadratura in , and Geometriae pars universalis in . In the first book particularly, he showed that the area of a circle can be obtained in the form of an infinite convergent series only, and hence inferred that the quadrature of the circle was impossible. http://www.advancesindifferenceequations.com/content/2013/1/100 In the second book, he attempted to write calculus systematically, which perhaps made the basis of Newton's fluxions. This book also contains series expansions of sin(x), cos(x), arcsin(x) and arccos(x); however, as we have seen earlier these expansions were known to Madhava. Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as . In early , he discovered Taylor's theorem (published by Taylor in ); however, he did not publish. Later in , he rediscovered Nilakanthan's arctangent series (). In his Vera circuli et hyperbolae quadratura of , Gregory tried to show that π was a transcendental number, but his attempt, though very interesting, was not successful. Huygens made detailed and rather biased criticisms of it.
. Pietro Mengoli (-) studied at the University of Bologna, and became a professor there in  for the next  years of his life. Besides proposing Basel problem, he proved that the harmonic series does not converge, established that the alternating harmonic series is equal to the natural logarithm of , published on the problem of squaring the circle, and provided a proof that Wallis' product () for π is correct.
. Gottfried Wilhelm von Leibniz (-) was a universal genius who won recognition in many fields -law, philosophy, religion, literature, politics, geology, metaphysics, alchemy, history and mathematics. He shares credit with Newton in developing calculus independently. He popularized and gave several mathematical symbols. Leibniz tried to reunite the Protestant and Catholic churches. He in binary arithmetic saw the image of Creation. He imagined that Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in the binary system of numeration. He communicated his idea to the Jesuit Grimaldi, who was the President of the Chinese tribunal for mathematics in the hopes that it would help convert to Christianity the Emperor of China, who was said to be very fond of the Sciences. Later Leibniz became an expert in the Sanskrit language and the culture of China. For calculating π , he developed a method without any reference to a circle. In , he also rediscovered Nilakanthan's arctangent series (), whose beauty he described by saying that Lord loves odd numbers. Leibniz even invented a calculating machine that could perform the four operations and extract roots.
. Father Adam Adamad Kochansky (-) was librarian of the Polish King John III. He was the first to utilize a steel spring for suspension of the pendulum of a clock. He used a new approximate geometric construction for π to obtain His method was later quoted in several geometrical textbooks. . Takebe Katahiro (-) also known as Takebe Kenko played a critical role in the development of a crude version of the calculus. He also created charts for trigonometric functions. He used polygon (just , sides) approximation and a numerical method which is essentially equivalent to the Romberg algorithm (rediscovered by Sigmund Romberg, -) to compute π to  digits. In , Takebe obtained power series expansion of (sin - x)  ,  years earlier than Euler. Around , essentially the same http://www.advancesindifferenceequations.com/content/2013/1/100 series was rediscovered by Oyama Shokei who used it to find the expansion The above expansion of π  was also given by Yamaji Nushizumi (-) around . . Abraham Sharp (-) was a mathematician and astronomer. In , he joined the Greenwich Royal Observatory and did notable work, improving instruments and showing great skill as a calculator. He also worked on geometry and improved logarithmic tables. In the supervision of Edmund Halley (-), he realized that by putting x =  in () (see ()) we lose the benefit of the powers x  , x  , x  , . . . , which tend to increase the rapidity of convergence for smaller values of x.
Sharp used () to calculated π to  decimal places out of which  digits are correct. In (), the th term is /( √  ·  ·   ), which is less than ., and hence we have at least  places correct after just  terms. It is believed that Madhava of Sangamagramma used the same series in the fourteenth century to compute the value of π correct to  decimal places.
. Seki Takakazu also known as Seki Kowa (-) is generally regarded as the greatest Japanese mathematician. He was a prolific writer, and a number of his publications are either transcripts of mathematics from Chinese into Japanese, or commentaries on certain works of well-known Chinese mathematicians. His interests in mathematics ranged recreational mathematics, magic squares and magic circles, solutions of higherorder and indeterminate equations, conditions for the existence of positive and negative roots of polynomials, and continued fractions. He discovered determinants ten years before Leibniz, and the Bernoulli numbers a year before Bernoulli. He used polygon of   sides and Richardson extrapolation (rediscovered by Alexander Craig Aitken, -) to compute π to  digits. Some authors believe that he also used the formula About . Oliver de Serres believed that by weighing a circle and a triangle equal to the equilateral triangle inscribed he had found that the circle was exactly double of the triangle, not being aware that this double is exactly the hexagon inscribed in the same circle. Thus, according to him π = . . William Jones (-), an obscure English writer, represented the ratio of the circumference of a circle to its diameter by π in his Synopsis Palmariorum Matheseos (New Introduction to the Mathematics). He used the letter π as an abbreviation for the Greek word perimetros (periphery) (of a circle with unit diameter). In his book, he published the value of π correct to  decimal places.
. John Machin (-) was a professor of astronomy at Gresham College, London. He also served as secretary of the Royal Society during -. Machin is best http://www.advancesindifferenceequations.com/content/2013/1/100 remembered for computing the value of π to  decimal places by using the formula which in view of () is the same as (   ) To establish (), we let tan θ = /, so that Thus, it follows that The proof of () also follows by comparing the angles in the identity (the idea originally goes back to Caspar Wessel (-) who presented his work in  to the Royal Danish Academy of Sciences) i.e., The series () certainly converges significantly faster than () and (). In fact, taking six terms of the first series and two terms of the second and paying attention to the remainders and round-off errors, we get the inequalities . < π < .. Thus, the value of π correct to seven decimals is .. Several other Machin-type formulas are known, e.g., For a long list of such type of formulas with a discussion of their relative merits in computational work, see Lehmer (). http://www.advancesindifferenceequations.com/content/2013/1/100 . Chapter  of Shu li jing yun (Collected Basic Principles of Mathematics), which was commissioned by the Emperor Kang Xi and edited by Mei Gucheng and He Guozong, gives π = ., which is correct to eight decimal places.
. Thomas Fantet de Lagny (-) was a French mathematician who is well known for his contributions to computational mathematics. He used the series () to determine the value of π up to  decimal places; however, only  are correct.
. Alexander Pope (-) was an English poet. He is the third-most frequently cited writer in The Oxford Dictionary of Quotations, after Shakespeare and Tennyson. In his Dunciad it is mentioned that 'The mad Mathesis, now, running round the circle, finds it square' . This explains the wild and fruitless attempts of squaring the circle.
. Sieur Malthulon (France) offered solutions to squaring the circle and to perpetual motion. He offered , crowns reward in legal form to anyone proving him wrong. Nicoli, who proved him wrong, collected the reward and abandoned it to the Hotel Dieu of Lyons. Later, the courts gave the money to the poor.
. Abraham De Moivre (-) was an intimate personal friend of Newton, and was elected an FRS of London in . In , he was appointed to the Commission set up by the Royal Society to solve the Newton-Leibniz dispute concerning which of them invented calculus first. He is best known for his memoir Doctrine of Chances: A method of calculating the probabilities of events in play, which was first printed in  and dedicated to Newton. In , he published his famous theorem (cos x + i sin x) n = cos nx + sin nx. In his Miscellanea Analytica published in , appears the formula for very large n, n! (πn) / e -n n n , which is known today as Stirling's formula. In , De Moivre used this formula to derive the normal curve as an approximation to the binomial.
. Leonhard Euler (-) was probably the most prolific mathematician who ever lived. He was born in Basel (Switzerland), and had the good fortune to be tutored one day a week in mathematics by a distinguished mathematician, Johann Bernoulli (-). Euler's energy and capacity for work were virtually boundless. His collected works form about  to  quarto-sized volumes and it is believed that much of his work has been lost. What is particularly astonishing is that Euler became virtually sightless in his right eye during the mid-s, and was blind for the last  years of his life, and this was one of the most productive periods. In , Mengoli asked for the precise summation of the infinite series ∞ n= n - . The series is approximately equal to . . . . . In the literature, this problem has been referred after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem. Basel problem appears in number theory, e.g., if two positive integers are selected at random and independently of each other, then the probability that they are relatively prime is ( ∞ n= n - ) - (R. Chartres, ). An integer that is not divisible by the square of any prime number is said to be square free. The probability that a randomly selected integer is square free is also ( ∞ n= n - ) - . Euler considered the function sin x/x, x =  which has the roots at ±nπ , n = , , . . . . Thus, it http://www.advancesindifferenceequations.com/content/2013/1/100 follows that Thus, on equating the coefficients of x  , we get which is the same as The above proof of Euler is based on manipulations that were not justified at the time, and it was not until  that he was able to produce a truly rigorous proof. It is interesting to note that () with x = π/ immediately gives Wallis' formula (). Today, several different proofs of () are known in the literature. Euler also established the following series: Later these statements. Since then with one exception, every statement has been settled in the sense Riemann expected. This exception is the famous Riemann hypothesis: that all the zeros of ζ (s) in the strip  ≤ σ ≤  lie on the central line σ = /. It stands today as the most important unsolved problem of mathematics, and perhaps the most difficult problem that the mind of man has ever conceived. . The letter π was first used by Euler in  in his Variae observationes circa series infinitas. Until that time, he had been using the letters p (), or c (). In , Christian Goldbach (-) also used π . After the publication of Euler's treatise: Introductio in Analysin Infinitorum (), π became a standard symbol, as was the case with other notations he adopted. In , Euler also showed that both e and e  are irrational and gave several continued fractions for e. In another paper, De variis modis circuli quadraturam numeris proxime exprimendi of , Euler derived the formulas and these give rise to any amount of relations for π ; for example, if x =  = y, and the odd numbers are substituted for a, b, c, . . . , we obtain The proof of () immediately follows by comparing the angles in the identity (p + q + i) p  + pq +  + iq = (p + q)  +  (p + i).
. Matsunaga Yoshisuke (died in ) was a prolific writer. In modern terms, he used the hypergeometric series for a = /, b = /, c = /, and x = /, i.e., the series to compute π correct to  digits. He also gave the following series: . The following expansion of π is due to Euler where the signs are determined following the rule: If the denominator is a prime of the form m -, the sign is positive; if the denominator is  or a prime of the form m + , the sign is negative; for composite numbers, the sign is equal to the product of signs of its factors. The following curious infinite product was also given by Euler: where the numerators are the odd primes and each denominator is the multiple of four nearest to the numerator.

. Jean Étienne Montucla (-) was an early French historian of mathematics.
He published an anonymous treatise entitled Histoire des récherches sur la quadrature du cercle, and in  the first part of his great work Histoire des mathématiques.

. Euler in his treatise De relatione inter ternas pluresve quantitates instituenda,
which was published ten years later, wrote 'It appears to be fairly certain that the periphery of a circle constitutes such a peculiar kind of transcendental quantities that it can in no way be compared with other quantities, either roots or other transcendentals' . This conjecture haunted mathematicians for  years. The following expansion is due to Euler: where y = x  /( + x  ). It converges rapidly.

. Georges Louis Leclerc (Comte of Buffon -) was a naturalist, mathemati-
cian, cosmologist and encyclopedic author. Suppose a number of parallel lines, distance a apart, are ruled on a horizontal plane, and suppose a homogeneous uniform rod of length < a is dropped at random onto the plane. Buffon showed that the probability that the rod will fall across one of the lines in the plane is given by p = ( /πa). In the literature, this problem is known as Buffon's needle problem. This was the earliest problem in geometric probability to be solved. By actually performing this experiment, a large number of times and noting the number of successful cases, we can compute an approximation for π .
. Johann Heinrich Lambert (-) was the first to introduce hyperbolic functions into trigonometry. He wrote landmark books on geometry, the theory of cartography, and perspective in art. He is also credited for expressing Newton's second law of motion in the notation of the differential calculus. Lambert used the properties of continued fractions to show that π is irrational. He published a more general result in .
Lambert also showed that the functions e x and tan x cannot assume rational values if x is http://www.advancesindifferenceequations.com/content/2013/1/100 a non-zero rational number. He also gave an interesting continued fraction for π , . Arima Yoriyuki (-) was a Japanese mathematician of the Edo period. He found the following rational approximation of π , which is correct to  digits π = ,,,, ,,,, .
. The French Academy of Sciences passed a resolution henceforth not to examine any more solutions of the problem of squaring the circle. In fact, it became necessary to protect its officials against the waste of time and energy involved in examining the efforts of circle squarers. A few years later, the Royal Society in London also banned consideration of any further proofs of squaring the circle. This decision of the Royal Society was described by Augustus De Morgan (-) about  years later as the official blow to circlesquarers.
. Charles Hutton (-) was an English mathematician. He wrote several mathematical texts. In , he was elected a fellow of the Royal Society of London. He suggested Machin's stratagem in the form however, he did not carry computations far enough. Euler also developed the formula (). . M. de Vausenville, one of the deluded individuals, brought an action against the French Academy of Sciences to recover a reward to which he felt himself entitled. It ought to be needless to say that there was no reward offered for squaring the circle.
. Euler used his expansion () to evaluate right terms of (), to calculate π to  decimal places in one hour! About . Franz Xaver Freiherr von Zach (-) discovered a manuscript by an unknown author in the Radcliffe Library, Oxford, which gives the correct value of π to  decimal places. Zach was elected a member of the Royal Swedish Academy of Sciences in , a Fellow of the Royal Society in , and an honorary member of the Hungarian http://www.advancesindifferenceequations.com/content/2013/1/100 Academy of Sciences in . Asteroid  Zachia and the crater Zach on the Moon are named after him.
. Baron Jurij Bartolomej Vega (Georg Vega -) was a Slovene mathematician, physicist and artillery officer. He wrote six scientific papers. The record of de Lagny of  digits seems to have stood until , when Vega, using a new series for the arctangent discovered by Euler in , calculated  decimal places ( correct). Vega's result showed that de Lagny's string of digits had a  instead of an  in the th decimal place. His article was not published until six years later, in  ( correct). Vega retained his record for  years until .
. Adrien-Marie Legendre (-) is remembered for Legendre functions, law of quadratic reciprocity for residues, standardizing weights and measures to the metric system, supervising the major task of producing logarithmic and trigonometric tables, least squares method of fitting a curve to the data available, proof of Fermat's last theorem for the exponent n = , Gauss-Legendre algorithm, Legendre's constant, Legendre's equation, Legendre polynomials, Legendre's conjecture, and Legendre transformation. The Legendre crater on the Moon is named after him. Legendre, in his Elements de Géometrie () used a slightly modified version of Lambert's argument to prove the irrationality of π more rigorously, and also gave a proof that π  is irrational. He writes: 'It is probable that the number π is not even contained among the algebraic irrationalities, i.e., that it cannot be the root of an algebraic equation with a finite number of terms, whose coefficients are rational. But, it seems to be very difficult to prove this strictly' .
. Ajima Naonobu (-), also known as Ajima Chokuyen, was a Japanese mathematician of the Edo period. The series he developed can be simplified as It is interesting to note that the above series follows from () by using an acceleration technique known in the literature as Euler's transform. It can also be derived from the Wallis product formula (). . Jean-Charles Callet (-) in his tables gave  ( correct) decimal digits of π .
. Lorenzo Mascheroni (-) was educated with the aim of becoming a priest and he was ordained at the age of . In , he calculated Euler's constant to  ( correct) decimal places. Lorenzo dedicated his book, Geometria del compasso, to Napoleon Bonaparte. In this work, he proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. However, it was proved earlier in  by the Danish mathematician Georg Mohr (-). He claimed that compasses are more accurate then those of a ruler.
. Karl Friedrich Gauss (-) was one of the greatest mathematicians of all time. Alexander von Humboldt (-), the famous traveler and amateur of the sciences, asked Pierre Simon de Laplace (-) who was the greatest mathematician in Germany, Laplace replied Johann Friedrich 'Pfaff ' (-). 'But what about Gauss' the astonished von Humboldt asked, as he was backing Gauss for the position of director at http://www.advancesindifferenceequations.com/content/2013/1/100 the Göttingen observatory. 'Oh' , said Laplace, 'Gauss is the greatest mathematician in the world' . Gauss suggested to his teacher Pfaff to study the sequences {x n } and {y n } generated by the recurrence relations In his reply, Pfaff showed that for any positive numbers x  and y  these sequences converge monotonically to a common limit given by () Pfaff 's letter was unpublished. In , Carl Wilhelm Borchardt (-) work was published in which he rediscovered this result which now bears his name. For this, it suffices to note that: . {x n } and {y n } converge monotonically to the same limit. . The ratio r n = x n /y n satisfies r  n+ = ( + r n )/. . If x  < y  , let θ = cos - r  . Then, s n =  n cos - r n = θ and c n =  n (x  ny  n ) = (x  y   ) are independent of n.
. lim n→∞ y n = lim n→∞ If y  < x  , we let θ = cosh - r  , and follow similarly. Now we let x n = /a n , y n = /b n , then () and () take the form a n+ = a n b n a n + b n , b n+ = a n+ b n , n ≥  (   ) and Clearly, the recurrence relations () are different from (). In fact, () minimize the count of arithmetic operations. In particular, if we let a  =  √ , b  = , then () in view of () converges to π .
In what follows, we let the constant c =  n (x  ny  n ) = (x  y   ), then we can uncouple () and (), respectively, to obtain He also estimated the value of π by using lattice theory and considering a lattice inside a large circle, but he did not pursue it further.

. Sakabe Kohan (-) developed the series
. Wada Yenzo Nei (known as Wada Yasushi, -) developed over one hundred infinite series expressing directly or indirectly π . One of his series can be written as . Malacarne of Italy published a geometric construction in Géométrique (Paris), which leads to the value of π less than .

. Karl Heinrich Schellbach (-) began with the relation
which is due to Giulio Carlo Fagnano dei Toschi (-), and used the logarithm expansion (), to obtain which immediately gives Nilakanthan series (). He also considered the relation and used the expansion (), to obtain, compare to (), a fast converging expansion

. William Baddeley in his work
Mechanical quadrature of the circle, London Mechanics' Magazine, August,  writes 'From a piece of carefully rolled sheet brass was cut out a circle  / inches in diameter, and a square  / inches in diameter. On weighing them, they were found to be exactly the same weight, which proves that, as each are of the same thickness, the surfaces must also be precisely similar. The rule, therefore, is that the square is to the circle as  to ' . We believe for the square it must be the side (not the diameter). Then it follows that π = ,/ = . . . . . http://www.advancesindifferenceequations.com/content/2013/1/100 . Joseph LaComme 'at a time when he could neither read nor write being desirous to ascertain what quantity of stones would be required to prove a circular reservoir he had constructed, consulted a mathematics professor. He was told that it was impossible to determine the full amount, as no one had yet found the exact relation between the circumference of a circle and its diameter. The well-sinker thereupon, full of self-confidence in his powers, applied himself to the celebrated problem and discovered the solution, which lead to π = / by mechanical process. He then taught himself to read and write, and managed to acquire some knowledge of arithmetic by which he verified his mechanical solution' . Joseph was honored for his profound discovery with several medals of the first class, bestowed by Parisian societies.
. William Rutherford (-) was an English mathematician. He calculated π to  places of which  were later found to be correct. For this, he employed Euler's formula and Madhava's series expansion (). . Johann Martin Zacharias Dase (-) was a calculating prodigy. At the age of , he gave exhibitions in Germany, Austria and England. His extraordinary calculating powers were timed by renowned mathematicians including Gauss. He multiplied ,, × ,, in  seconds; two -digit numbers in  minutes; two -digit numbers in  minutes; and two -digit numbers in  hours  minutes. In , he made acquaintance with Viennese mathematician L.K. Schulz von Strasznicky (-) who suggested him to apply his powers to scientific purposes. When he was , Strasznicky taught him the use of the formula and asked him to calculate π . In two months, he carried the approximation to  places of decimals, of which  are correct. He next calculated a -digit logarithm table of the first ,, numbers; he did this in his off-time from  to , when occupied by the Prussian survey. His next contribution of two years was the compilation of hyperbolic This series can be written as to calculate π to  decimal places, but only  are correct. In , he factored the sixth Fermat number as   +  = ,,,, × ,. Clausen also gave a new method of factorising numbers. . Jacob de Gelder (-) a mathematical ideologist proposed a geometric construction which gives π correct to  decimal places. His method is based on the fact that π = / =  +   /(  +   ). Gelder's result was published in .
. Joseph Liouville (-) was a highly respected professor at the Collége de France in Paris, and the founder and for  years the editor of the Journal des Mathématiques Pures et Appliquées. His ingenious theory of fractional differentiation answered the long-standing question of what reasonable meaning can be assigned to the symbol d n y/dx n when n is not a positive integer. In , Liouville showed that e is not a root of any quadratic equation with integral coefficients. This led him to conjecture that e is transcendental. In , Liouville showed, by using continued fractions, that there are an infinite number of transcendental numbers, a result which had previously been suspected but had not been proved. He produced the first examples of real numbers that are provably transcendent. One of these is . Lehmann correctly calculated  decimal places of π . For this, he used Euler's formula . Rutherford obtained  correct decimal places. . William Shanks (-) was a British amateur mathematician. He used Machin formula () to calculate π to  decimal places. He was assisted by Rutherford in checking first  digits.
-. Richter in  published  digits ( correct), and in  (after his death in )  decimal places.
. James Smith published the value of π as  / and argued that it is exact and correct. He attempted to bring it before the British Association for the Advancement of Science. Interestingly, even De Morgan and Hamilton could not convince him for his mistake.

.
Lawrence Sluter Benson published about  pamphlets on the area of the circle, three volumes on philosophic essays, and one on geometry The Elements of Euclid and Legendre. He demonstrated that the area of the circle is equal to R  , or the http://www.advancesindifferenceequations.com/content/2013/1/100 arithmetical square between the inscribed and circumscribed squares. According to him √  = . . . . is the ratio between the diameter of a circle and the perimeter of its equivalent square. The ratio between the diameter and circumference, he believed, is not a function of the area of the circle. He accepted the value of π = . . . . . . S.M. Drach proved that the circumference of a circle can be obtained as follows: From thrice diameter, deduct /, and /,, of a diameter, and add % to the result, i.e., which gives π = .. . Cyrus Pitt Grosvenor (-) was an American anti-slavery Baptist minister. In his retirement, he worked on the problem of squaring the circle. He described his method in a pamphlet titled The circle squared, New York: Square the diameter of the circle; multiply the square by ; extract the square root of the product; from the root subtract the diameter of the circle; square the remainder; multiply this square by four fifths; subtract the square from the diameter of the circle, i.e., which gives π = . . . . . . Augustus De Morgan (-) was born in Madura (India), but his family moved to England when he was seven months old. He lost the sight of his right eye shortly after birth. He was an extremely prolific writer. He wrote more than , articles for more than  periodicals. De Morgan also wrote textbooks on many subjects, including logic, probability, calculus and algebra. In , he was a co-founder of the London Mathematical Society and became its first President. His book A Budget of Paradoxes of  pages, which was edited and published by his wife in , is an entertaining text. This book contains the names of  writers on π . In this work, De Morgan reviewed the works of  of these writers, bringing the subject down to . He once remarked that it is easier to square the circle then to get round a mathematician. He was the first to point out that in the decimal expansion of π one should expect each of the  digits appear uniformly, i.e., roughly one out of ten digits should be a , etc.
. Asaph Hall (-) was an astronomer. He published the results of an experiment in random sampling that Hall had convinced his friend, Captain O.C. Fox, to perform when Fox was recovering from a wound received at the Second Battle of Bull Run. The experiment was based on Buffon's needle problem. After throwing his needles eleven hundred times, Fox was able to derive π .. This work is considered as a very early documentation use of random sampling (which Nicholas Constantine Metropolis (-) named as the Monte Carlo method during the Manhattan Project of World War II).
. Charles Hermite (-) in  was appointed to a professorship at the Sorbonne, where he trained a whole generation of well-known French mathematicians. He was strongly attracted to number theory and analysis, and his favorite subject was elliptic functions, where these two fields touch in many remarkable ways. His proof of the transcendence of e was high point in his career. http://www.advancesindifferenceequations.com/content/2013/1/100 -. William Shanks again used Machin formula () to calculate π to  decimal places (published in the Proceedings of the Royal Society, London), but only  decimal places are correct. For this, he used mechanical desk calculator and worked for almost  years. For a long time, this remained the most fabulous piece of calculation ever performed. In the Palais de la Découverte (a science museum in Paris), there is a circular room known as the 'pi room' . On its wall are inscribed these  digits of π . The digits are large wooden characters attached to the dome-like ceiling. Shanks also calculated e and the Euler-Mascheroni constant γ to many decimal places. He published a table of primes up to , and found the natural logarithms of , ,  and  to  places.
. Tseng Chi-Hung (died in ) finds  digits of π in a month. For this, he used the formula ().
. John A. Parker in his book The Quadrature of the Circle. Containing Demonstrations of the Errors of Geometers in Finding Approximations in Use published by John Wiley & Sons, New York claims that π = ,/, exactly. He exclaims, 'all the serial and algebraic formula in the world, or even geometrical demonstration, if it be subjected to any error whatever, cannot overthrow the ratio of circumference to diameter which I have established' . He praises Metius (lived in the sixteenth century) for using the ratio /. His book also contains practical questions on the quadrature applied to the astronomical circles.
. Alick Carrick proposed in his book, The Secret of the Circle, its Area Ascertained, the value of π as  /. . Pliny Earle Chase (-) was a scientist, mathematician, and educator who mainly contributed to the fields of astronomy, electromagnetism and cryptography. In his pamphlet, Approximate Quadrature of the Circle, he used a geometric construction to obtain π = . . . . . . Carl Louis Ferdinand von Lindemann (-) worked on non-Euclidean geometry. He followed the method of Hermite to show that π is also transcendental. His result showed at last that the age-old problem of squaring the circle by a ruler-and-compass construction is impossible. Lindemann's paper runs to  pages of tough mathematics. Karl Wilhelm Weierstrass (-), the apostle of mathematical rigor, simplified the proof of Lindemann's theorem somewhat in , and it was further simplified in later years by renowned mathematicians (Stieltjes, Hurwitz, Hilbert, and others). The interested reader is referred to the comparatively easy version given by Hobson. Nonetheless, there are still some amateur mathematicians who do not understand the significance of this result, and futilely look for techniques to square the circle. Next, Lindemann spent several years to provide the proof of Fermat's Last Theorem, which is unfortunately wrong. He also worked on projective geometry, Abelian functions and developed a method of solving equations of any degree using transcendental functions.
. Sylvester Clark Gould (-) was the editor of Notes and Queries, Manchester, New Hampshire. He compiled the bibliography entitled What is the Value of Pi. It contains  titles and gives the result of  authors. In this work the diagram  claims that π =  ,/, exactly. . A writer announced in the New York Tribune the rediscovery of a long-lost secret that gives . as the exact value of π . This announcement caused considerable discussion, and even near the beginning of the twentieth century . had its advocates as against the value /. http://www.advancesindifferenceequations.com/content/2013/1/100 . Fredrik Carl Mülertz Störmer (-) was a mathematician and physicist, known for his work in number theory. He gave the following Machin-like formulas for calculating π and . In the State of Indiana, the House of Representatives unanimously passed the Bill No.  (known as the 'π 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 on one side. . . ' (π = .). The author of the bill was a physician, Edwin J. Goodman (-), M.D., of Solitude, Posey County, Indiana, and it was introduced in the Indiana House on  January , by Mr. Taylor I. Record, representative from Posey County. Edwin offered this contribution as a free gift for the sole use of the State of Indiana (the others would evidently have to pay royalties). Copies of the bill are preserved in the Archives Division of the Indiana State Library. The bill was sent to the Senate for approval. Fortunately, during the House's debate on the bill, Purdue University Mathematics Professor Clarence Abiathar Waldo (-) was present. When Professor Waldo informed the Indiana Senate of the 'merits' of the bill, the Senate, after some ridicule at the expense of their colleagues, indefinitely postponed voting on the bill and let it die.
. H.S. Uhler used Machin's formula () to compute π to  decimal places. . Mario Lazzarini an Italian mathematician performed the Buffon's needle experiment. Tossing a needle , times, he obtained the well-known estimate / for π . Although it is an impressive observation, but suspiciously good. In fact, statisticians Sir Maurice George Kendall (-) and Patrick Alfred Pierce Moran (-) FRS have commented that one can do better to cut out a large circle and use a tape to measure to find its circumference and diameter. On the same theme of phoney experiments, Gridgeman, in , pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = ., throwing it twice, and hitting a line once. His estimate for π was thus given by  × ./π = / from which he got the highly creditable value of π = .. Of course, he was not being serious.
. Duarte used Machin's formula () to compute π to  decimal places. . Various mnemonic devices have been given for remembering the decimal digits of π . The most common type of mnemonic is the word-length mnemonic in which the number of letters in each word corresponds to a digit, for example, How I wish I could calculate pi (by C. Heckman), May I have a large container of coffee (by Martin Gardner), and How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics (by Sir James Jeans), respectively, give π to seven, eight, and fifteen decimal places. Adam C. Orr in Literary Digest, vol.  (), p. published the following poem which gives π to  decimal places: http://www.advancesindifferenceequations.com/content/2013/1/100 Now I, even I, would celebrate In rhymes inapt, the great Immortal Syracusan, rivaled nevermore Who in his wondrous lore, Passed on before Left men his guidance, How to circles mensurate.
Several other such poems not only in English, but almost in every language including Albanian, Bulgarian, Czech, Dutch, French, German, Italian, Latin, Polish, Portuguese, Romanian, Spanish and Swedish are known. However, there is a problem with this type of mnemonic, namely, how to represent the digit zero. Fortunately, a zero does not occur in π until the thirty-second place. Several people have come up with ingenious methods of overcoming this, most commonly using a ten-letter word to represent zero. In other cases, a certain piece of punctuation is used to indicate a naught. Michael Keith (with such similar understanding) in his work Circle digits: a self-referential story, Mathematical Intelligencer, vol.  (), -, wrote an interesting story which gives first  decimals of π .
. Ernest William Hobson (-) was Sadleirian Professor at the University of Cambridge from  to . His  work on real analysis was very influential in England. In his book, Squaring the circle: A History of the Problem, he used a geometrical construction to obtain π = . . . . . . Srinivasa Ramanujan (-) was a famous mathematical prodigy. He collaborated with Hardy for five years, proving significant theorems about the number of partitions of integers. Ramanujan also made important contributions to number theory and also worked on continued fractions, infinite series and elliptic functions. In , he became the youngest Fellow of the Royal Society. According to Hardy, 'the limitations of Ramanujan's knowledge were as startling as its profundity' . Here was a man who could workout modular equations and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta-function, and the dominant terms of the many of the most famous problems in the analytic theory of numbers; and he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was. Ramanujan considered mathematics and religion to be linked. He said, 'an equation for me has no meaning unless it expresses a thought of God' . He was endowed with an astounding memory and remembered the idiosyncrasies of the first , integers to such an extent that each number became like a personal friend to him. Once Hardy went to see Ramanujan when he was in a nursing home and remarked that he had traveled in a taxi with a rather dull number, viz ,, Ramanujan exclaimed, 'No, Hardy, , is a very interesting number. It is the smallest number that can be expressed as the sum of two cubes viz , =   +   =   +   , and the next such number is very large'. His life can be summed up in his own words, 'I really love my subject' . His  paper on 'Modulus functions and approximation to π ' contains several new innovative empirical formulas and geometrical constructions for approximating π . One of the remarkable formulas for its elegance and inherent mathematical depth http://www.advancesindifferenceequations.com/content/2013/1/100 It has been used to compute π to a level of accuracy, never attained earlier. Each additional term of the series adds roughly  digits. He also developed the series The first series has the property that it can be used to compute the second block of k (binary) digits in the decimal expansion of π without calculating the first k digits. The following mysterious approximation which approximates π to  correct decimal places is also due to Ramanujan . Edmund Georg Hermann (Yehezkel) Landau (-) was a child prodigy. In , he gave a simpler proof of the prime number theorem. His masterpiece of  was a treatise Handbuch der Lehre von der Verteilung der Primzahlen a two volume work giving the first systematic presentation of analytic number theory. Landau wrote over  papers on number theory, which had a major influence on the development of the subject. Despite his outstanding talents as both a teacher and researcher, Landau annoyed many of his colleagues at Göttingen. He started criticizing privately, and often publicly, their results. Landau in his work defined π/ as the value of x between  and  for which cos x vanishes. One cannot believe this definition was used, at least as an excuse, for a racial attack on http://www.advancesindifferenceequations.com/content/2013/1/100

. T.M.P. Hughes in his work
Landau. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen. Ludwig Georg Elias Moses Bieberbach (-) famous for his conjecture, explained 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 un-German 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' . Hardy replied immediately to Bieberbach about the consequences of this un-German 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. ' . A Cleveland businessman Carl Theodore Heisel published a book Mathematical and Geometrical Demonstrations in which he announced the grand discovery that π was exactly equal to /, a value that the Egyptians had used some , years ago. Substituting this value for calculations of areas and circumferences of circles with diameters , , . . . up to , he obtained numbers which showed consistency of circumference and area, 'thereby furnishing incontrovertible proof of the exact truth' of his ratio. He also rejected decimal fractions as inexact (whereas ratios of integers as exact and scientific), and extracted roots of negative numbers thus: twentieth century, also took part. In , The ENIAC became obsolete, and it was dismembered and moved to the Smithsonian Institution as a museum piece.
. Konrad Knopp gave the following two expansions of π : . Kurt Mahler (-) showed that π is not a Liouville number: A real number x is called a Liouville number if for every positive integer n, there exist integers p and q with q >  and such that A Liouville number can thus be approximated 'quite closely' by a sequence of rational numbers. In , Liouville showed that all Liouville numbers are transcendental. . S.C. Nicholson and J. Jeenel programmed NORC (Naval Ordnance Research Calculator) at Dahlgren, Virginia to compute π to , decimals. For this, they used Machin's formula (). The run took only  minutes.
. John Gurland established that for all positive integers n, (   ) March . G.E. Felton used the Ferranti Pegasus computer to find , decimal places of π in  hours. The program was based on Klingenstierna's formula However, a subsequent check revealed that a machine error had occurred, so that 'only' , decimal places were correct. The run was therefore repeated in May , but the correction was not published. January . Francois Genuys programmed an IBM  at the Paris Data Processing Center. He used Machin's type formula (). It yielded , decimal places of π in  hour and  minutes.
July . Genuys programmed an IBM  at the Commissariat á l'Energie Atomique in Paris to compute π to , decimal places. He used Machin's type formula (). It took  hours and  minutes.
July . Daniel Shanks (-) and William Shanks used Störmer's formula () on an IBM  (at the IBM Data Processing Center, New York) to compute π to , digits, of which the first , digits were published by photographically reproducing the print-out with , digits per page. The time required for this computation was  hours and  minutes. They also checked the calculations by using Gauss' formula (), which required  hours and  minutes.
. Machin's formula () was also the basis of a program run on an IBM  at the London Data Center in July , which resulted in , decimal places and required only  minutes running time. http://www.advancesindifferenceequations.com/content/2013/1/100 February . Jean Guilloud and J. Filliatre used an IBM  at the Commissariat á l'Energie Atomique in Paris to obtain an approximation of π extending to , decimal places on a STRETCH computer. For this, they used Störmer's and Gauss' formulas () and (). It took almost  hours.
February . Guilloud and M. Dichampt used CDC (Control Data Corporation)  in Paris to approximate π to , decimal places. For this, they used Störmer's and Gauss' formulas () and (). The computer that churned out half a million digits needed only  hours and  minutes (plus  hour and  minutes to convert that final result from binary to decimal notation).
. In the Putnam Competition, the first problem was This integral was known to Mahler in the mid-s, and has later appeared in several exams. It is also discussed by Borwein, Bailey . Guilloud with Martine Bouyer (Paris) used formulas () and () on a CDC  to compute π to ,, digits. The run time required for this computation was  hours and  minutes, of which  hour  minutes was used to convert the final result from binary to decimal. Results of statistical tests, which generally support the conjecture that π is simply normal (in , Félix Édouard Justin Émil Borel (-) defined: A real number a is simply normal in base b if in its representation in base b all digits occur, in an asymptotic sense, equally often) were also performed.
. Louis Comtet developed the following Euler's type expansion of π : . Richard Brent and Eugene Salamin independently discovered an algorithm which is based on an arithmetic-geometric mean and modifies slightly Gauss-Legendre algorithm. Set a  = , b  = / √  and s  = /. For k = , , , . . . compute Then p k converges quadratically to π , i.e., each iteration doubles the number of accurate digits. In fact, successive iterations must produce , , , , , , ,  and  correct digits of π . The twenty-fifth iteration must produce  million correct decimal digits of π . . Kazunori Miyoshi and Kazuhika Nakayama of the University of Tsukuba, Japan calculated π to ,, significant figures in . hours on a FACOM M- computer. They used Klingenstierna's formula () and checked their result with Machin's formula ().
. Guilloud computed ,, decimal digits of π . . Rajan Srinivasan Mahadevan (born ) recited from memory the first , digits of π . This secured him a place in the  Guinness Book of World Records, and he has been featured on Larry King Live and Reader's Digest.
. Kikuo Takano (-) was a Japanese poet and mathematician. He developed the following Machin-like formula for calculating π : . Yoshiaki Tamura on MELCOM II computed ,, decimal places of π . For this, he used the Salamin-Brent algorithm ().
. Yoshiaki Tamura and Yasumasa Kanada (born , life-long 'pi digit-hunter' , set the record  of the past  times) on HITAC M-H computed ,, decimal places of π . For this, they used the Salamin-Brent algorithm ().
. Yoshiaki Tamura and Yasumasa Kanada on HITAC M-H computed ,, decimal places of π . For this, they used the Salamin-Brent algorithm ().
October . Yasumasa Kanada, Yoshiaki Tamura, Sayaka Yoshino and Yasunori Ushiro on HITAC S-/ computed ,, decimal places of π . For this, they used the Salamin-Brent algorithm (). In this work to gather evidence that π is simply normal, they also performed statistical analysis. It showed expected behavior. In the first ten million digits, the frequencies for each ten digits are ,; ,; ,,; ,; ,,; ,,; ,; ,,; ,; and ,,. Further, the rate at which the relative frequencies approach / agrees with theory. As an example, for the digit  relative frequencies in the first  i , i = , , , , , , ,  digits are , ., ., ., ., ., ., which seem to be approaching / at rate predicted by the probability theory for random digits, i.e., a speed approximately proportional to / √ n. But this is far from a formal proof of simple normalcy perhaps for a proof the current mathematics is not sufficiently developed. In spite of the fact that the digits of π pass statistical tests for randomness, π contains some sequences of digits that, to some, may appear non-random, such as Feynman point, which is a sequence of six consecutive s that begins at the nd decimal place. A number is said to be normal if all blocks of digits of the same length occur with equal frequency. Mathematicians expect π to be normal, so that every pattern possible eventually will occur in the digits of π . http://www.advancesindifferenceequations.com/content/2013/1/100 . Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura on HITAC M-H computed ,, decimal places of π . For this, they used the Salamin-Brent algorithm ().
. Jonathan Borwein and Peter Borwein gave the following algorithm. Set x  = √ , y  =  and α  =  + √ . Iterate Then α k converges to π quartically. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits of π . . Morris Newman and Daniel Shanks proved the following: Set Then a k converges quartically to /π , i.e., each iteration approximately quadruples the number of correct digits. . The following is not an identity, but is correct to over  billion digits . Carl Sagan in his novel deals with the theme of contact between humanity and a more technologically advanced, extraterrestrial life form. He suggests that the creator of the universe buried a message deep within the digits of π . http://www.advancesindifferenceequations.com/content/2013/1/100 January . David H. Bailey used Borweins' algorithms () and () on CRAY- to compute ,, decimal places of π .
Then /a k converges cubically to π , i.e., each iteration approximately triples the number of correct digits. Among the several other known iterative schemes, we list the following two which are easy to implement on a computer: Set a  = / and s  = ( √  -). Iterate x n+ = /s n -, , a n+ = s  n a n - n s  n -  + s n s  n -s n +  , Then a k converges quintically to /π , i.e., each iteration approximately quintuples the number of correct digits, and  < a n -/π <  ·  n · e - n π . http://www.advancesindifferenceequations.com/content/2013/1/100 Set a  = /, r  = ( √  -)/ and s  = (r   ) / . Iterate t n+ =  + r n , a n+ = w n+ a n +  n- (w n+ ), Then a k converges nonically to /π , i.e., each iteration approximately multiplies the number of correct digits by nine. . David Bailey, Peter Borwein and Simon Plouffe developed the following formula (known as BBP formula) to compute the nth hexadecimal digit (base ) of π without having the previous n - digits

. Jonathan Borwein and Peter Borwein developed the series
To show the validity of (), for any k < , we have Substituting u = √ x in equation (), we obtain   u - u  -u  + u - du =   u u  - du -  u - u  -u +  du = π.
The discovery of this formula came as a surprise. For centuries, it had been assumed that there was no way to compute the nth digit of π without calculating all of the preceding n - digits. Since this discovery, many such formulas for other irrational numbers have been discovered. Such formulas have been called as spigot algorithms because, like water dripping from a spigot, they produce digits that are not reused after they are calculated. . Simon Plouffe discovered an algorithm for the computation of π in any base. Later he expressed regrets for having shared credit for his discovery of this formula with Bailey and Borwein.
March . David and Gregory Chudnovsky used a home made parallel computer m zero to obtain ,,, decimal places of π . For this, they used series (). They said 'we are looking for the appearance of some rules that will distinguish the digits of π from other numbers, i.e., if someone were to give you a million digits from somewhere in π , could you tell it was from π? The digits of π form the most nearly perfect random sequence of digits that has ever been discovered. However, each digit appears to be orderly. If a single digit in π were to be changed anywhere between here and infinity, the resulting number would no longer be π , it would be garbage. Around the three-hundred-millionth decimal place of π , the digits go -eight eights pop up in a row. Does this mean anything? It appears to be random noise. Later, ten sixes erupt: . What does this mean? Apparently nothing, only more noise. Somewhere past the half-million mark appears the string . It is an accident, as it were. We do not have a good, clear, crystallized idea of randomness. It cannot be that π is truly random. Actually, a truly random sequence of numbers has not yet been discovered' .
July . Yasumasa Kanada and Daisuke Takahashi on HITACHI SR (, CPU) computed ,,, decimal places of π . The computation tool just over  hours, at an average rate of nearly , digits per second. For this, they used algorithms () and ().
. Fabrice Bellard developed the following formula: × -  m +  - m +  +   m +  -  m +  -  m +  -  m +  +  m +  , which can used to compute the nth digit of π in base . It is about % faster then ().
The following exotic formula is also due to him: π =  , . J. Munkhammar gave the following formula which is related to Viéte's (): which as a recurrence relation can be written as π = lim n→∞  n+ a n , where a  = √ , and a n =   a n-  +  - -  a n-   .
Another closely related formula is . Robert Palais believes that the notation π is wrongly used right from the beginning. According to him, some suitable symbol (now popular as tau τ ) must have been used for π . He justifies his claim by giving several formulas where τ appears naturally rather than just π . For some people, June , is Tau's Day and they celebrate.
November . Yasumasa Kanada used Machin-like formulas () and () to compute the value of π to ,,,, decimal places. The calculation took more than  hours on  nodes of a HITACHI SR/MPP supercomputer. The work was done at the Department of Information Science at the University of Tokyo. For this, he used arctangent formulas () and ().
. Daniel Tammet, at age , recited , decimal places of π , scoring the European record. For an audience at the Museum of the History of Science in Oxford, he said these numbers aloud for  hours and  minutes. Unfortunately, he made his first mistake at position , and did not correct this error immediately and without outside help, but only after he was told that there was a mistake.
Several other integral formulas of this type are known, here we give the following: simple concise Matlab program would do the job wherever π /any other irrational number is involved. . Syamal K. Sen, Ravi P. Agarwal and Raffela Pavani have provided, using Matlab, the best possible rational bounds bracketing π /any irrational number with absolute error and the time complexity involved. Any better bounds are impossible. In these rational bounds, either the lower bound or the upper bound will always be the absolute best rational approximation. The absolute error computed provides the overall error bounds in an error-free computational environment involving π /any other irrational number.
August . Daisuke Takahashi et al. used a massive parallel computer called the TK Tsukuba System to compute π to ,,,, decimal places in  hours  minutes. For this, they used algorithms () and ().
December . Fabrice Bellard used Chudnovsky brothers series () to compute ,,,,, i.e., . trillion decimal places of π in  days. For this, he used a single desktop PC, costing less than $,.
August . Shigeru Kondo and Alexander J. Yee used Chudnovsky brothers series () to compute ,,,,, i.e.,  trillion decimal places of π in  days. For this, they used a server-class machine running dual Intel Xeons, equipped with  GB of RAM.
. Michael Keith used , digits of π to establish a new form of constrained writing, where the word lengths are required to represent the digits of π . His book contains a collection of poetry, short stories, a play, a movie script, crossword puzzles and other surprises.
. Syamal K. Sen and Ravi P. Agarwal in their monograph systematically organized their work of  and  on π and other irrational numbers. They also included several examples to illustrate the importance of their findings. http://www.advancesindifferenceequations.com/content/2013/1/100

Conclusions
No number system can capture π exactly. We are deeply and almost completely involved in the conventional decimal number system in representing any real quantity. This is not the only number system for the representation. There are other number systems such as binary, octal, hexadecimal, binary-coded decimal, negative radix, p-adic and modular number systems. If the circumference of a circle is exactly represented, then its diameter will not have exact representation and vice versa.
Reading the mathematicians in pre-computer days. An important focus of this paper is that the reader besides, however, knowing the usual chronology of the events in the life of π , could get a feel and also read how the mind of a mathematician has been working when he ponders over π either independently without much knowledge/concern of what has been done in the past or with considerable knowledge of the work done by his predecessors. Hyper-computers (  flops) of  were completely non-existence and even beyond the imagination of all the mathematicians/scientists until almost the mid-twentieth century. Also, publication machinery was too poor until the beginning of the twentieth century. Consequently, all the work on π that has been carried out during thousands of years prior to the twentieth century was not a monotonic improvement in the π value as well as in the exploration of its wonderful character. Many have worked on π standalone while others have contributed with some prior knowledge of the earlier work. All of them were severely handicapped due to the non-existence of today's ultra-high speed computers. They entirely depended on their ingenuity and on whatever negligible computing device they had. It is really interesting under this environment to read these scientists/mathematicians and realize how fortunate we are in the gigantic computer age. All that has been done during the last  years (-) amounts to much more than what has been achieved during the past several millennia.
Matlab is well-suited to check/evaluate merits of all past π formulas. Widely used userfriendly Matlab that needs no formal programming knowledge along with the vpa (variable precision arithmetic) and format long g commands can be used to easily and readily check all that has been done during the past several thousand years and possibly appreciate the inherent intellectual import of the bygone scientists (having practically no computing device) and their expected pitfalls, bias and incorrect beliefs.
Checking exactness of billions of digits of π is difficult. Are all the billions of digit of π computed % error-free? We are familiar with the age-old proverb that 'To err is human (living being)' . Maybe a new proverb 'Not to err is computer (non-living being)' can be taken as true in the modern computer age. Here, 'err' means mistake. The arithmetic operations, particularly subtraction operations of two nearly equal numbers, involved in a formula could be sometimes error introducer. However, different computers with different formulas used to compute π would help verification and obviate possible error in computation.
Computing nth decimal digit exactly always without preceding digits seems yet an open computational problem. While probabilistically one may determine the nth digit of π without computing the preceding n - digits, obtaining nth digit exactly (correctly) always for any n does not seem to be possible without a large precision. It seems yet an open computational problem that needs exploration. Thus, formulas such as () seem more of theoretical/academic interest than of practical usage as of now. http://www.advancesindifferenceequations.com/content/2013/1/100 PI for testing performance and stability of a computer. Super PI is a computer program that calculates π to a specified number of digits after decimal point up to a maximum of  million digits. It uses the Gauss-Legendre algorithm and is a Windows port of the program used by Yasumasa Kanada in  to compute π to   digits. Super PI is used by many overclockers to test the performance and stability of their computers. Overclocking is the process of making a computer run faster than the clock frequency specified by the manufacturer by modifying system parameters.