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The Man Who Invented Pi

In 1706 a little-known mathematics teacher named William Jones first used a symbol to represent the platonic concept of pi , an ideal that in numerical terms can be approached, but never reached.

The history of the constant ratio of the circumference to the diameter of any circle is as old as man's desire to measure; whereas the symbol for this ratio known today as π ( pi ) dates from the early 18th century. Before this the ratio had been awkwardly referred to in medieval Latin as: quantitas in quam cum multiflicetur diameter, proveniet circumferencia (the quantity which, when the diameter is multiplied by it, yields the circumference).

It is widely believed that the great Swiss-born mathematician Leonhard Euler (1707-83) introduced the symbol π into common use. In fact it was first used in print in its modern sense in 1706 a year before Euler's birth by a self-taught mathematics teacher William Jones (1675-1749) in his second book Synopsis Palmariorum Matheseos , or A New Introduction to the Mathematics based on his teaching notes.

Before the appearance of the symbol π, approximations such as 22/7 and 355/113 had also been used to express the ratio, which may have given the impression that it was a rational number. Though he did not prove it, Jones believed that π was an irrational number: an infinite, non-repeating sequence of digits that could never totally be expressed in numerical form. In Synopsis he wrote: '... the exact proportion between the diameter and the circumference can never be expressed in numbers...'. Consequently, a symbol was required to represent an ideal that can be approached but never reached. For this Jones recognised that only a pure platonic symbol would suffice.

The symbol π had been used in the previous century in a significantly different way by the rector and mathematician, William Oughtred (c. 1575-1 660), in his book Clavis Mathematicae (first published in 1631). Oughtred used π to represent the circumference of a given circle, so that his π varied according to the circle's diameter, rather than representing the constant we know today. The circumference of a circle was known in those days as the 'periphery', hence the Greek equivalent 'π' of our letter 'π'. Jones's use of π was an important philosophical step which Oughtred had failed to make even though he had introduced other mathematical symbols, such as :: for proportion and 'x' as the symbol for multiplication.

On Oughtred's death in 1660 some books and papers from his fine mathematical library were acquired by the mathematician John Collins (1625-83), from whom they would eventually pass to Jones.

The irrationality of π was not proved until 1761 by Johann Lambert (1728-77), then in 1882 Ferdinand Lindemann (1852-1939) proved that π was a non-algebraic irrational number, a transcendental number (one which is not a solution of an algebraic equation, of any degree, with rational coefficients). The discovery that there are two types of irrational numbers, however, does not detract from Jones's achievement in recognising that the ratio of the circumference to the diameter could not be expressed as a rational number.

Beyond his first use of the symbol π Jones is of interest because of his connection to a number of key mathematical, scientific and political characters of the 18th century. He was also responsible for developing one of the greatest scientific libraries and mathematical archives in the country which remained in the hands of the Macclesfield family, his patrons, for nearly 300 years.

Though Jones ended his life as part of the mathematical establishment, his origins were modest. He was born on a small farm on Anglesey in about 1675. His only formal education was at the local charity school where he showed mathematical aptitude and it was arranged for him to work in a merchant's counting house in London. Later he sailed to the West Indies and became interested in navigation; he then went on to be a mathematics master on a man-of-war. He was present at the battle of Vigo in October 1702 when the English successfully intercepted the Spanish treasure fleet as it was returning to the port in north-west Spain under French escort. While the victorious seamen went ashore in search of silver and the spoils of war, for Jones, according to an 1807 memoir by Baron Teignmouth, '... literary treasures were the sole plunder that he coveted.'

On his return to England Jones left the Navy and began to teach mathematics in London, probably initially in coffee houses where for a small fee customers could listen to a lecture. He also published his first book, A New Compendium of the Whole Art of Practical Navigation (1702). Not long after this Jones became tutor to Philip Yorke, later 1st Earl of Hardwicke (1690-1764), who became lord chancellor and provided an invaluable source of introductions for his tutor.

It was probably around 1706 that Jones first came to Isaac Newton's attention when he published Synopsis, in which he explained Newton's methods for calculus as well as other mathematical innovations. In 1708 Jones was able to acquire Collins's extensive library and archive, which contained several of Newton's letters and papers written in the 1670s. These would prove of great interest to Jones and useful to his reputation.

Born half a century apart, Collins and Jones never met, yet history will forever link them because of the library and mathematical archive that Collins started and Jones continued, arising from their shared passion for collecting books. The son of an impoverished minister, Collins was apprenticed to a bookseller. Essentially self-taught like Jones, he had also gone to sea and learned navigation. On his return to London he had earned his living as a teacher and an accountant. He held several increasingly lucrative posts and was adept at disentangling intricate accounts.

Collins's modest ambition had been to open a bookshop, but he was unable to accumulate enough capital. In 1667, however, he was elected to the Royal Society of which he became an indispensable member, assisting the official secretary Henry Oldenburg on mathematical subjects. Collins corresponded with Newton and with many of the leading English and foreign mathematicians of the day, drafting mathematical notes on behalf of the Society.

When Jones applied for the mastership of Christ's Hospital Mathematical School in 1709 he carried with him testimonials from Edmund Halley and Newton. In spite of these he was turned down. However Jones's former pupil, Philip Yorke, had by now embarked on his legal career and introduced his tutor to Sir Thomas Parker (1667-1732), a successful lawyer who was on his way to becoming the next lord chief justice in the following year. Jones joined his household and became tutor to his only son, George (c.1697-1764). This was the start of his life-long connection with the Parker family.

Around the time that Jones bought Collins's library and archive, Newton and the German mathematician Gottfried Leibniz (1646-1716) were in dispute over who invented calculus first. In Collins's mathematical papers, Jones had found a transcript of one of Newton's earliest treatments of calculus, De Analyst (1669), which in 1711 he arranged to have published. It had previously been circulated only privately. President of the Royal Society since 1703, Newton was reluctant to have his work published and jealously guarded his intellectual property. However, he recognised an ally in Jones.

In 1712 Jones joined the committee set up by the Royal Society to determine priority for the invention of calculus. Jones made the Collins papers with Newton's correspondence on calculus available to the committee and the resulting report on the dispute, published later that year, Commercium Epistolicum , was based largely upon them. Though anonymous, Commercium Epistolicum was edited by Newton himself and could hardly be viewed as impartial. Unsurprisingly it came down on Newton's side. (Today it is considered that both Newton and Leibniz discovered calculus independently though Leibniz's notation is superior to Newton's and is the one now in common use.)

By 1712 Jones was firmly positioned among the mathematical establishment. In 1718 his patron Sir Thomas Parker was made lord chancellor and in 1721 was ennobled as Earl of Macclesfield. By this time he had purchased Shirburn estate and castle for the then vast sum of £18,350. Shirburn castle became a home too for Jones who was, by then, almost a family member. Besides the law, Parker had a scholarly interest in many subjects including science and mathematics and was a generous patron of the arts as well as the sciences. He was influential in the appointment of Halley as astronomer royal in 1721.

But there was an obverse side to the first earl's character. It seems that together with his great abilities and ambition there was also a dangerous lust for wealth. He was accused of selling chancery masterships to the highest bidder and of allowing suitors' funds held in trust to be misused. Parker resigned as lord chancellor in 1725 but he was nevertheless impeached. His punishment was a fine of £30,000 and he was forced to spend six weeks in the Tower of London before the necessary money was raised to pay the fine. Some of his assets were sold and his name was struck from the roll of privy councillors but he did not have to forfeit Shirburn which remains in the Macclesfield family to this day. Some dignity was restored when in 1727 he was one of the pallbearers at Newton's funeral.

Thomas's son, George Parker, became an MP for Wallingford in 1722 and spent much of his time at Shirburn where, with Jones's guidance, he added to the library and archive that Jones had brought with him. George Parker developed an interest in astronomy and with the help of a friend, the astronomer James Bradley (who became the third Astronomer Royal in 1742 on the death of Halley), he built an astronomical observatory at Shirburn.

By 1718 Jones was dividing his time mainly between Shirburn and Tibbald's Court, near Red Lion Square, London. Among the many influential mathematicians, astronomers and natural philosophers he corresponded with was Roger Cotes (1682-1716), the first Plumian Professor of Astronomy at Cambridge and considered by many to be the most talented British mathematician of his generation after Newton. He had been entrusted with the revisions for the publication of the second edition of Newton's Principia .

Jones acted as a conduit between Newton and Cotes when relations between the two became strained. He clearly had influence and considerable tact. In one letter Cotes wrote to Jones: 'I must beg your assistance and management in an affair, which I cannot so properly undertake myself ...'. This was the delicate matter of suggesting to Newton an improvement in one of his methods. Newton had a difficult personality and had to be handled carefully. This Jones was able to do. The second, amended edition of Principia was published in 1713 to great acclaim.

Newton was a towering eminence over most of the period and many among the scientific community lived under his shadow. Jones also had an extensive correspondence with the astronomer and mathematician, John Machin (c.1686-1771), who served as secretary to the Royal Society for nearly 30 years from 1718. He was also on the Society's committee to investigate the invention of calculus. Professor of astronomy at Gresham College for nearly 40 years, Machin worked on lunar theory and considered himself an expert on the subject. In one letter to Jones, Machin used fanciful language to complain about Newton's lunar theory:

... she (the moon) has informed me that he (Newton) has abused her throughout the whole course of her life, giving out that she is guilty of such irregularities and enormities in all her ways and proceedings that no man alive is able to find where she is at any time.

He then went on to write that he, Machin, knew the moon's whereabouts and would therefore be able to claim the £10,000 which the 'Lord Treasurer' was offering for the discovery of longitude at sea; because his lunar theory would improve the accuracy of lunar tables.

Though Machin did not receive the reward, his lunar theory as described in Laws of the moon's motion according to gravity was appended to the 1729 English edition of Principia after Newton's death.

Machin had also worked on a series for the ratio of the circumference to the diameter which converged fairly rapidly. The result of his calculation was printed in Jones's 1706 book, 'true to above a 100 places; as computed by the accurate and ready pen of the truly ingenious Mr John Machin...'. Machin performed this by using an infinite series whose sum converged to π. In mathematical terms this means that no matter how many terms are summed there is always a difference, however small, between that sum and the value of the irrational number, π. In the infinite series, which Machin used, the terms alternate between being positive and negative so that the sum is alternately lower or higher than π.

Jones also had correspondents abroad; one of particular interest was the Quaker scholar James Logan (1674-1751) who lived in America. Logan had been born in Ireland and was invited by William Penn, the Quaker leader and founder of Pennsylvania, to be his secretary. He prospered there and eventually bought a plantation, Stenton, where he retired in his early fifties to pursue his interests, including mathematics and botany. His own library of over 30,000 books was one of the most outstanding of the 18th century in America and was bequeathed to the city of Philadelphia.

In 1732 Logan wrote to Jones about an invention by, 'a young man here ... of an excellent natural genius'. This was Thomas Godfrey (1704-49), a glazier, who in October 1730 had invented an instrument that could be accurately used at sea because it had a single half-mirrored sight that lined up a reflected image of the sun with the horizon. Alternatively any two astronomical objects, for instance, the moon and a star could be lined up by moving a rotatable arm containing the mirror and reading off the angle from the scale. This meant that movement of a ship would not interfere with the angular measurement as both object and image would move together. It was an ingenious instrument. Logan considered that it could be used to find longitude at sea by the lunar method. The instrument is what we now know as Hadley's Quadrant, although it is in fact an octant. The attribution of this important invention was claimed both by America and by England. The English astronomer John Hadley (1682-1744) had made one of these instruments in the summer of 1730 and sent an account to the Royal Society the following May.

Logan had sent a personal letter describing Godfreys invention to Halley, then President of the Royal Society, addressing him as 'Esteemed Friend'. It was a friendly communication as well as a scientific one and was not read to the Royal Society, as was customary. Logan asked Jones to make some enquiry about the omission. Jones subsequently raised the subject with the Society in January 1734 and Godfrey's claims to be the inventor of the instrument, though not the first, were established.

Some years later in 1736 Jones wrote to Logan, apologising for not having replied sooner, saying that:

... my affairs are such as require my constant application, and take up my mind so much that I have little, or no leasur (sic) to think of anything else: even the mathematics. I have scarce thought of it these 18 years past, and am now almost a stranger to all improvements made that way.

But there are letters in Jones's correspondence dating from after that time that are mathematical in subject. Perhaps he did not want to encourage Logan to send him further discoveries. Logan was a tireless correspondent and it appears that he wrote many more letters to Jones than Jones answered.

There were certainly other things on Jones's mind. Like many other men of science, Jones was intrigued with the problem of longitude and he wrote letters to the Royal Society on the subject of clocks keeping accurate time as the temperature changed.

He served as a council member of the Society and became its vice-president in 1749. His income was boosted by sinecures organised by his former pupils: he was made Secretary of the Peace through the influence of Hardwicke and Deputy Teller to the Exchequer with George Parker's help. Nevertheless, he also experienced financial crisis on more than one occasion when his bank collapsed, a frequent occurrence in those days.

Jones married a second time in 1731 to Mary Nix, 30 years his junior and they had three children. He was elected a Governor of the Foundling Hospital in 1747 when George Parker was vice-president. It was Parker who commissioned Hogarth's portrait of Jones. Although Jones looked impressive in this portrait, he is reported to have been 'a little short faced Welshman, and used to treat his mathematical friends with a great deal of roughness and freedom'. Even so, as we have seen, he knew how to be tactful when necessary and could show great kindness.

After he died in 1749, aged 74, it was reportedly said by John Robertson, a clerk and librarian to the Royal Society, that he 'died in better circumstances than usually falls to the lot of mathematicians'. His one surviving son, also called William, was only three years old at the time. Known as Oriental' Jones, he excelled as a linguist, philologist and expert in Hindu Law and was duly knighted.

In 1750 George Parker wrote a paper which was read to the Royal Society entitled Remarks upon the Solar and Lunar years . Parker was a principal proponent for the adoption of the Gregorian calendar and the change in 1752 of the new year from March 25th to January 1st. One might consider the revision of the calendar as part of William Jones's scientific legacy. The same year Parker was elected president of the Royal Society, a position he held until his death.

In his will, Jones left his 'study of books' to George Parker 'as a testimony of my acknowledgement of the many marks of his favour which I have received'. The scientific books Parker inherited from Jones, together with the archive of papers, remained in the library at Shirburn. Access to them had been severely restricted though it was acknowledged that they represented the most important collection of their kind in private hands. In 2000 the archive of letters and papers was offered to Cambridge University Library who purchased it for £6,370,000 with the aid of a grant from the Heritage Lottery Fund. The Macclesfield Library was finally sold at Sotheby's in 2005 in six massive sales that have replenished libraries throughout the world.

In his lifetime, Jones's ability to retain his patrons was important and he served them well. From a historical perspective though, Jones gave much more to the Macclesfields than he ever received from them and, in doing so, he left a great intellectual legacy to the world.

Patricia Rothman is Honorary Research Fellow in the Department of Mathematics at University College, London

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The surprising history of π

Actually, it’s not all that greek to me, meghan bartels • march 14, 2016.

Pie pi is clearly the tastiest way to represent pi, but what’s the oldest? [Image credit: Kristin Brenemen | CC BY-NC-ND 2.0 ]

The tyranny of π, jonathan chang • june 29, 2012, q+a with mark umile, ryan f. mandelbaum • march 14, 2016.

The Greeks get a lot of credit for a lot of things, but that doesn’t always mean they deserve it. Take π, that magical number that turns a circle’s diameter into its circumference: It’s designated by a Greek letter and that Archimedes guy estimated it with some polygons , so it must be Greek, right?

Sure, this is (a much later, printed Latin translation of) Archimedes calculating π — pretty accurately, no less, but let’s not get too excited. [Image credit: Biblioteca Europea di Informazione e Cultura | CC BY-NC-ND 3.0 ]

For most of history, π wasn’t really represented by any particular symbol at all. Like Archimedes, many mathematicians just wrote out what they were talking about, using whatever variation of “the ratio of the circumference of a circle to its diameter” they happened to prefer. Otherwise, they just used the current numerical approximation. Three was a popular simplification, but many estimates were actually quite close to the value we know today.

And even when mathematicians finally decided it would be handy to have a standard symbol, there were a few other contenders to fill the role of what became π.

John Wallis uses a square to represent the ratio of a circle’s area to the square of its radius. [Image credit: Archive.org | public domain]

In which William Jones can’t quite decide if he wants to invent π. [Image courtesy Archives and Special Collections, Bangor University ]

William Jones gets down to business, saving future mathematicians from representing a number with a clunky phrase. [Image courtesy Archives and Special Collections, Bangor University ]

But even once π was the undisputed symbol for the ratio between a circle’s circumference and diameter, there was still the tricky matter of actually remembering the number it represented. One popular approach was to compose mnemonic devices, often structured so that the number of letters in each word matched each digit of π, a strategy called pilish . There are dozens of examples in languages from Albanian to Swedish.

English speakers may particularly enjoy this charmer, published in the magazine of the Royal Observatory at Greenwich. The editor suspected it was submitted by a famous astronomer of the time, who noted it had been tailored for an American audience. “How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics.”

Now π is second nature to us, but next time you find yourself impatient with calculations, spare a thought for the mathematicians who spent millennia doing math with words instead of symbols and raise your glass to π!

About the Author

Meghan Bartels

Meghan Bartels graduated from Georgetown University with a major in classics and a minor in biology. After college, she worked at a small environmental book publisher, where she learned that writing about science is fun when you get to use sentences that include both nouns and verbs. She also enjoys learning about history, drinking tea, and cheering on the Georgetown men’s basketball team.

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A Brief History of Pi

That the ratio of the circumference of a circle to its diameter is constant has been known to humanity since ancient times; yet, even today, despite 2000 years of thought, theories, calculations and proofs, π’s precise value remains elusive.

Ancient Civilizations

By the 17th century B.C., the Babylonians had a relatively advanced knowledge of mathematics, that they memorialized into complicated tables that expressed squares, fractions, square and cube roots, reciprocal pairs and even algebraic, linear and quadratic equations.

It should come as no surprise, then, that these math whizzes had also discerned an estimate of π at:

Contemporaneous with the Babylonians, the Egyptians were also making great strides with mathematics, and are believed to have developed the first full-fledged base 10 number system.

The oldest evidence of π in Egypt is found in the Rhind Papyrus, which dates from about 1650 B.C. Together with instructions for multiplication and division, and evidence of prime numbers, fractions and even some linear equations, the Egyptian π was calculated as

When the Hebrews were building the Temple of Solomon around 950 B.C., they recorded its specifications, including that of a large brass casting as described in I Kings 7:23: “ Then he made the molten sea; it was made with a circular rim, and measured 10 cubits across, five in height and thirty in circumference.”

  Note that the ratio between the circumference and the diameter is 3. Not terribly precise, but also not bad, considering they had only emerged from the wilderness a few centuries before.

  The Greeks greatly advanced the study of mathematics, and particularly the field of geometry. One of their earliest quests, dating back to at least the 5th century B.C., was to “square the circle” – create a square with exactly the same area as a circle. Although many tried, none were quite able to accomplish the feat, although the reason why was not explained for another 2000 years.

In any event, by the 3rd century B.C., Archimedes of Syracuse, the great engineer and inventor, devised the first known theoretical calculation of π as:

About 400 years later, another Greek, Ptolemy, further refined the estimate of π using the chords of a circle with a 360-sided polygon to obtain:

Dating back to 2000 B.C. and built on a 10 based, place value system, Chinese mathematics were well developed by the 3rd century A.D. when Liu Hiu, who also developed a type of early calculus, created an algorithm to calculate π to five correct decimal places.

Two-hundred years later, Zu Chongzhi calculated to six decimal places, and demonstrated the following:

Middle Ages

Working in the 9th century A.D., Muhammad Al-Khwarizmi , widely credited with creating two of algebra’s most fundamental methods (balancing and reducing), the adoption of the Hindu numbering system (1-9, with the addition of a 0) and the inspiration for the words algebra and algorithm , is said to have calculated π accurately to four decimal places.

Several hundred years later, in the 15th century A.D., Jamshid al-Kashiintroduced his Treatise on the Circumference in which he calculated 2 π to 16 decimal places.

From al-Kashi’s time through to the 18th century, developments related to pi generally were limited to producing ever more precise approximations. About 1600, Ludolph Van Ceulen calculated it to 35 decimal places, while in 1701, John Machin, who is credited with creating better methods for approximating π, was able to produce 100 digits.

In 1768, Johann Heinrich Lambert proved that pi is an irrational number, meaning it is a real number that cannot be written as a quotient of integers (recall Archimedes’ calculation, where π exists between two quotients of integers, but isn’t defined by one).

There was a π lull again, until finally, in the late 19th century, two more interesting things happened: in 1873, William Shanks correctly calculated pi to 527 places (he actually produced 707, but the last 180 were wrong), and in 1882, Carl Louis Ferdinand von Lindemann proved, in Über die Zahl , that π is transcendental , meaning:

Pi transcends the power of algebra to display it in its totality. It can’t be expressed in any finite series of arithmetical or algebraic operations. Using a fixed-size font, it can’t be written on a piece of paper as big as the universe.

  Because he proved pi’s transcendence, Lindemann also proved, once and for all, that there was no way one could “square the circle.”

Americans (well, Hoosiers)

In the 19th century, not everyone kept up on the latest in the world of mathematics. This must have been the case with Indiana amateur mathematician Edwin J. Goodwin. In 1896, he had so convinced himself that he had, in fact, found a way to “square the circle,” that he talked a representative of the Indiana House into introducing a bill (to become a law) that his value of pi was correct.

Luckily, before the Indiana legislature got too far down that road, a visiting Purdue University professor informed the esteemed body that it was impossible to square the circle, and, in fact, Goodwin’s “proof” was based on two errors, most pertinent to this article, the error that

If you liked this article, you might also enjoy our new popular podcast, The BrainFood Show ( iTunes , Spotify , Google Play Music , Feed ), as well as:

• The Truth About Einstein and Whether He Failed at Mathematics as a Child
• The Horse that Could Do Math: The Unintentional Clever Hans Hoax
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Bonus Fact:

• The mathematical volume of a pizza is pizza.  How does that work you say?  Well if z = radius of the pizza and a = the height then Π * radius 2 * height = Pi * z * z * a = Pizza.
• 15 Most Famous Transcendental Numbers
• Chinese Mathematics
• Egyptian Mathematics
• Greek Mathematics
• A History of Pi
• The History of the Indiana Pi Bill
• Islamic Mathematics
• I Kings, chapter 7
• Johann Heinrich Lambert
• Carl Louis Ferdinand von Lindemann
• Pi Day: History of Pi (Exploratorium)
• Sumerian/Babylonian Mathematics
• Zu Chongzhi

• Featured Facts

26 comments

Not one of Indiana’s proudest moments. Can we just forget this ever happened and move on? Huh?

2 for the road ——————-

Fibonacci sequence – 1 1 2 3 5 8 13 21 34 55 89 144 ……. v 1.618

The Fibonacci sequence is named after Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics. By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber Abaci began the sequence with F1 = 1, without an initial 0. Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, … . Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings,[such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.

Origins The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody. In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers; the number of patterns that are m short syllables long is the Fibonacci number Fm + 1. Susantha Goonatilake writes that the development of the Fibonacci sequence “is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150)”. Parmanand Singh cites Pingala’s cryptic formula misrau cha (“the two are mixed”) and cites scholars who interpret it in context as saying that the cases for m beats (Fm+1) is obtained by adding a [S] to Fm cases and [L] to the Fm−1 cases. He dates Pingala before 450 BC. –

Redshift increases with distance ——————————————–

The law states that the greater the distance between any two galaxies, the greater their relative speed of separation.

This discovery was the first observational support for the Big Bang theory which had been proposed by Georges Lemaître in 1927 – a Belgian Roman Catholic priest, astronomer and professor of physics at the Université catholique de Louvain.

Earlier, in 1917, Albert Einstein had found that his newly developed theory of general relativity indicated that the universe must be either expanding or contracting. Unable to believe what his own equations were telling him, Einstein introduced a cosmological constant (a “fudge factor”) to the equations to avoid this “problem”.

In the 1930s, Hubble was involved in determining the distribution of galaxies and spatial curvature. These data seemed to indicate that the universe was flat and homogeneous, but there was a deviation from flatness at large redshifts. When Einstein learned of Hubble’s redshifts, he immediately realized that the expansion predicted by General Relativity must be real, and in later life he said that changing his equations was “the biggest blunder of [his] life.” In fact, Einstein apparently once visited Hubble and tried to convince him that the universe was expanding. The Higgs boson is named after Peter Higgs, one of six physicists who, in 1964, proposed the mechanism that suggested the existence of such a particle. Although Higgs’s name has come to be associated with this theory, several researchers between about 1960 and 1972 each independently developed different parts of it. In mainstream media the Higgs boson has often been called the “God particle”, from a 1993 book on the topic; the nickname is strongly disliked by many physicists, including Higgs, who regard it as inappropriate sensationalism. In 2013 Peter Higgs and François Englert were awarded the Nobel Prize in Physics for their discovery.

Other references: ———————- Stephen William Hawking – born 8 January 1942 The Higgs boson or Higgs particle Edwin Powell Hubble (November 20, 1889 – September 28, 1953) The Hubble Space Telescope (HST) Kepler is a space observatory launched by NASA The James Webb Space Telescope (JWST) The European Organization for Nuclear Research – known as CERN The International Space Station (ISS)

– Wikipedia, the free encyclopedia

52 ——

Fifty-two is the 6th Bell number and a decagonal number. It is an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is never the answer to the equation x – φ(x).

The first few untouchable numbers are: 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, … (sequence A005114 in OEIS)

The number 5 is believed to be the only odd untouchable number, but this has not been proven: it would follow from a slightly stronger version of the Goldbach conjecture. Thus it appears that besides 2 and 5, all untouchable numbers are composite numbers. No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors. There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős.

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Descriptive essay on pi – it’s meaning and history | student example for free.

Pi is the most known mathematical constant. As was mentioned in the introduction, it equals the circumference of a circle divided by the diameter. According to LiveScience, “No matter how large or small a circle is, pi will always work out to be the same number. That number equals approximately 3.14, but it’s a little more complicated than that” (Hom, Elaine J.). Pi is also noted as an irrational number, which basically points to the fact that is a real number that can be shown as a simple fraction. In fact, it is what mathematicians call an “infinite decimal,” as pi’s digits go on forever. Also, there are no repeating patterns in pi.

Since pi’s digits go on forever, mathematicians and scientists often approximate it. According to Wonderopolis, “For most purposes, pi can be approximated as 3.14159. Some people even shorten it to 3.14, which is why Pi Day is celebrated on March 14 (3/14)” (“What Is Pi?”). What is interesting is that computers have calculated pi to over three trillion digits. Some people even remember 1000s of digits of pi as a mental exercise.

But where did this number come from? Surprisingly, this formula has been in use for over 4000 years. The first people to discover pi was the Babylonians. According to Exploratorium, “The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for π, which is a closer approximation. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π” (“A Brief History of Pi (π)”). However, the first calculation of pi was completed by Archimedes (287–212 BC). Yet, he knew he did not find the value of pi, but rather an approximation of its value, with him stating that pi was between 3 1/7 and 3 10/71.

Pi has many roles. In mathematics, “Most geometry students first encounter pi when they study circles and learn that the area of a circle is equal to pi times the square of the length of the radius.” It is also used widely in trigonometry. Formulae concerning circles, spheres, or ellipses commonly use pi. Here are some common formulae that use it: The circumference of a circle with radius r is 2πr. The area of a circle with radius r is πr2. The volume of a sphere with radius r is 4/3πr3. The surface area of a sphere with radius r is 4πr2. (A Guide Book to Mathematics)

In physics, pi is also commonplace. According to LiveScience, “Pi also appears in the physics that describes waves, such as ripples of light and sound. It even enters into the equation that defines how precisely we can know the state of the universe, known as Heisenberg’s uncertainty principle. A river’s windiness is determined by its “meandering ratio,” or the ratio of the river’s actual length to the distance from its source to its mouth as the crow flies” (Wolchover, Natalie). Therefore, pi has many applications in physics as well.

Pi is surprisingly in popular culture. There is an even a day celebrated for this mathematical constant. However, most do not know that it was first discovered by the ancient Babylonians and refined by Archimedes. Also, many people do not know that it is an irrational number that equals the circumference of a circle divided by the diameter. This special formula has many applications in mathematics, physics, and even more areas.

Works Cited

Hom, Elaine J. “What Is Pi?” LiveScience, Purch, 19 Oct. 2018, www.livescience.com/29197-what-is-pi.html.

“What Is Pi?” Wonderopolis, wonderopolis.org/wonder/what-is-pi.

“A Brief History of Pi (π).” Exploratorium, 5 Mar. 2019, www.exploratorium.edu/pi/history-of-pi.

A Guide Book to Mathematics. Springer Verlag, 2012.

Wolchover, Natalie. “What Makes Pi So Special?” LiveScience, Purch, 9 Aug. 2012, www.livescience.com/34132-what-makes-pi-special.html.

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A Brief History of Pi (π) and Pie (🥧)

Meghan Marino, Associate Manager, Field Marketing, Wiley

March 14, 2024

Happy Pi (π) Day! π isn’t just a random number; it’s the key to understanding circles. By comparing the circumference of a circle to its diameter, we arrive at the magic number of 3.141592653589(…), better known as – yes, you guessed it – pi. 

But this mathematical constant isn’t just celebrated through numbers; it’s often paired with its homophone: pie. So, this year, we want to look at a history where mathematics and delicious treats collide – with materials from Wiley Digital Archives!

How old is Pi? Pi has been around a long time – close to 4,000 years in fact. Ancient Babylonian tablets from circa 1900 BC calculated pi’s value at 3, while one tablet refined that number to an even closer 3.125. Flashing forward to around 1650 BC, the Ancient Egyptians found an approximate value of 3.16 before the first close approximation was done by the ancient Greek mathematician Archimedes (287-212 BC), who declared pi’s value somewhere around 22/7, or 3.142857. The brilliantly precise Chinese mathematician Zu Chongzhi took π even further during the 5th century, refining its value between 3.1415926 and 3.1415927.

Our understanding of this magical number has been a long, collaborative work in progress involving mathematicians from all over the world. By 1750, pi was given its current symbol, expressed by a value with over 100 digits. Even in the 21st century, researchers continue to expand its meaning. The current record sits at a value of 62,831,853,071,796 digits , a number that took Swiss mathematicians over 108 days to obtain.

Archimedes in the Archives While many archival documents cover Archimedes, perhaps the oldest in our archives is dated from 1599. “Scholarum mathematicarum libri unus et triginta” or “Thirty-one books of mathematical schools” speaks of Archimedes’ first theorem on the dimension of a circle, among his other mathematical accomplishments. 

View the collection : Pierre de la Ramee, Lazarus Schoner, RCP Library, M.D.XCIX, The Royal College of Physicians archive . (Available via trial or institutional access)

Zu Chongzhi in the Archives From the times of ancient China and beyond, a manuscript highlighting accomplished scholars writes that Zu Chongzhi was “the first person who accurately calculated the value of π to seven decimal places.”

View the collection: Qin Shi, Essays, literary works, and biographical papers, 1992, The Royal Antropological Institute of Great Britain and Ireland archive. (Available via trial or institutional access)

A Bite-Size History of Pie Before refrigerators, pie served as a tasty means to preserve food. So, how long have we enjoyed sweet and savory treats like apple pie and pizza pie? Well, the first written recipe was for a rye-crusted goat cheese and honey pie, and is credited to the Romans by the American Pie Council . But the true age of these treats is debatable; there is evidence of a recipe for chicken pie on a carved Ancient Egyptian tablet dating back to over 2000 BC.

Pie in the Archives Pizza has become so ingrained in numerous cultures that it’s hard to picture a time when it was considered exotic. But a book on Naples from 1901 shares a perspective completely in awe of the food that many of us love and can’t picture life without.

“’Pizza’ may be seen in every street in Naples,” writes British author Arthur H. Norway. “It is a kind of biscuit, crisp and flavoured with cheese, recognizable at a glance by the little fish, like whitebait, which are embedded in its brown surface, dusted over with green chopped herbs.” 

View the collection : Arthur H. Norway, Monographs, 1901, Royal Geographical Society (with IBG) archive . (Available via trial or institutional access)

In a newspaper article titled "America is a Bad Tourist Trap," a disgruntled British visitor expresses their dissatisfaction with American delicacies. The bold statement is made that “’homemade blueberry pie’ in the roadside diner is revealed at every trial as the same old tasteless slice of purple glue,” among many other quirky insights from a disappointed Brit.

View the collection : David Holden, William Buller Fagg Collection, n.d., the Royal Anthropological Institute of Great Britain and Ireland archive . (Available via trial or institutional access)

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If you’re curious about other stories from the archives, we invite you to dive into the fascinating life of Sophie Germain, a self-taught mathematician who, under a male pseudonym, submitted college assignments and became the first woman to receive an award in the field of mathematics.

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• DOI: 10.2307/2005843
• Corpus ID: 61661894

A History of Pi

• P. Beckmann
• Published 1970
• History, Mathematics

128 Citations

Historia matheseos. early development stage history of mathematics, historiography, the great l's of french mathematics, anniversary of notation for number π, lessons from the greeks and computers, in search of a place in history for mathematics: a lecture series in south africa, the quest for pi, ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi, real analysis, a riemann sum approach to buffon’s needle, related papers.

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Review: A History of Pi, by Petr Beckmann

A History of Pi , published in 1971, is about two things. The first is pi, the number, and our history struggling with its baffling qualities. The second is fascism and, more broadly, ignorance. Beckmann holds pi like a mirror up to humanity and sees stunning genius and highly confident idiots and thugs. The last line of the book notes that there are more of the latter than the former.

The joining of these two topics makes this book really fun to read. The math is dense, though–denser than most pop math books these days. You can skip what you don’t understand (or don’t want to bother with), but some parts of this story can only be told in mathematics, and if you want to appreciate, say, Newton’s contributions to our understanding of pi, or Euler’s, you need to dig into some math that involves infinite sums and integrals. Even the ancient stuff can be daunting. This book is intended for a numerate audience. If equations make you uncomfortable, I wouldn’t recommend Beckmann. If they don’t, on the other hand, the math is quite thrilling, and elegantly presented. Euler’s derivation of

1/1+1/4+1/9+1/16+… 1/(n squared) + … = (pi squared)/6

is breathtakingly elegant, as is Lindemann’s proof of the transcendence of pi. But, again, I’m afraid much of this will read as opaque to a lay audience.

On the other hand, his uninhibited judgments and thorough historical research make the book strangely thrilling. He compares Archimedes’ writing to Aristotle’s: the first is “science,” the second “prattle.” I’ve never heard Aristotle described as a prattler before, but it’s kind of liberating to hear Beckmann say so. The Romans come off far worse; they are “thugs” to Beckmann, early fascists and forerunners to the Nazis and the Soviets. Their use of pi bears out his judgment: they used 3 and 1/8 as an approximation of pi, which was about 2000 years behind the times. As the Romans empire expands, one soldier comes across as thinker doodling shapes in the sand during the sacking of Syracuse.

“Do not touch my circles!” said the thinker to the thug. Thereupon the thug became enraged, drew his sword and slew the thinker.

The name of the thug is forgotten.

The name of the thinker was Archimedes.

Here is the drama that Beckmann sees played out throughout history: the genius is dispatched by the moron.

There was a series of math books by Lilian Lieber that share the same sense of mathematics (and serious thought, more generally) as being a kind of bulwark against fascism. There’s something I love in this idea. There’s also a delicacy to the whole project: mathematics is the flower and tyranny the backhoe… and yet, if enough of us possess the thoroughness of thought to appreciate the art and beauty of math, perhaps we can prevent that backhoe from ever getting started on its destructive path.

Beckmann is perhaps a bit more cynical than Lieber, but his love and respect for the great thoughts of the giants of mathematics is evident. Science and math march forward, and Beckmann is on its side, even as he bridles against those who push towards darkness. All in all, the book was a pleasure to read, and I recommend it to anyone who won’t be scared off by the denseness of the mathematics.

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Pi: The Next Generation

A Sourcebook on the Recent History of Pi and Its Computation

• © 2016
• David H. Bailey 0 ,
• Jonathan M. Borwein 1

Berkeley, USA

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Centre for Computer Assisted Research Ma, University of Newcastle, Newcastle, Australia

Presents amazing techniques for computing digits of pi as well as high-tech techniques for analyzing pi

Brief synopses precede each contribution containing a summary of its content and a short key word list indicating how the content relates to others in the collection

Presents a modern collection of papers dealing with pi and associated topics in mathematics and computer science

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About this book

This book contains a compendium of 25 papers published since the 1970s dealing with pi and associated topics of mathematics and computer science.  The collection begins with a Foreword by Bruce Berndt. Each contribution is preceded by a brief summary of its content as well as a short key word list indicating how the content relates to others in the collection. The volume includes articles on actual computations of pi, articles on mathematical questions related to pi (e.g., “Is pi normal?”), articles presenting new and often amazing techniques for computing digits of pi (e.g., the “BBP” algorithm for pi, which permits one to compute an arbitrary binary digit of pi without needing to compute any of the digits that came before), papers presenting important fundamental mathematical results relating to pi, and papers presenting new, high-tech techniques for analyzing pi (i.e., new graphical techniques that permit one to visually see if pi and other numbers are “normal”).  

This volume is a companion to Pi: A Source Book whose third edition released in 2004.  The present collection begins with 2 papers from 1976, published by Eugene Salamin and Richard Brent, which describe “quadratically convergent” algorithms for pi and other basic mathematical functions, derived from some mathematical work of Gauss. Bailey and Borwein hold that these two papers constitute the beginning of the modern era of computational mathematics.  This time period (1970s) also corresponds with the introduction of high-performance computer systems (supercomputers), which since that time have increased relentlessly in power, by approximately a factor of 100,000,000, advancing roughly at the same rate as Moore’s Law of semiconductor technology.  This book may be of interest to a wide range of mathematical readers; some articles cover more advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students.

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Table of contents (25 chapters)

Front matter, computation of π using arithmetic-geometric mean (1976).

• Eugene Salamin

Fast multiple-precision evaluation of elementary functions (1976)

• Richard P. Brent

The arithmetic-geometric mean of Gauss (1984)

• David A. Cox

The arithmetic-geometric mean and fast computation of elementary functions (1984)

• J. M. Borwein, P. B. Borwein

A simplified version of the fast algorithms of Brent and Salamin (1985)

• D. J. Newman

Is pi normal? (1985)

The computation of π to 29,360,000 decimal digits using borweins quartically convergent algorithm (1988).

David H. Bailey

Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary (1988)

• Gert Almkvist, Bruce Berndt

Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation (1988)

• Yasumasa Kanada

Ramanujan and pi (1988)

• Jonathan M. Borwein, Peter B. Borwein

Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi (1989)

• Jonathan M. Borwein, Peter B. Borwein, David H. Bailey

Pi, Euler numbers, and asymptotic expansions (1989)

• Jonathan M. Borwein, Peter B. Borwein, Karl Dilcher

A spigot algorithm for the digits of π (1995)

• Stanley Rabinowitz, Stan Wagon

On the rapid computation of various polylogarithmic constants (1997)

• David H. Bailey, Peter B. Borwein, Simon Plouffe

Similarities in irrationality proofs for π, ln 2, ζ(2), and ζ(3) (2001)

• Dirk Huylebrouck

Unbounded spigot algorithms for the digits of pi (2006)

• Jeremy Gibbons

Mathematics by experiment: Plausible reasoning in the 21st Century (2008)

• David H. Bailey, Jonathan M. Borwein

Approximations to π derived from integrals with nonnegative integrands (2009)

• Stephen K. Lucas

Ramanujan’s series for 1/π: A survey (2009)

• Nayandeep Deka Baruah, Bruce C. Berndt, Heng Huat Chan

Authors and Affiliations

Jonathan M. Borwein