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The Man Who Invented Pi

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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.

Patricia Rothman | Published in History Today Volume 59 Issue 7 July 2009

William Jones, mathematician from Wales, 1740

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.

Source : www.historytoday.com

Pi Day: History of Pi

A Brief History of Pi (π) Pi (π) has been known for almost 4000 years—but even if we calculated the number of seconds in those 4000 years and calculated π to that number of places, we would still only be approximating its actual value. Here’s a brief history of finding π. The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which

A Brief History of Pi (π)

Pi (π) has been known for almost 4000 years—but even if we calculated the number of seconds in those 4000 years and calculated π to that number of places, we would still only be approximating its actual value. Here’s a brief history of finding π.

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 Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π.

The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world. Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of π but only an approximation within those limits. In this way, Archimedes showed that π is between 3 1/7 and 3 10/71.

A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for π, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.

An eighteenth-century French mathematician named Georges Buffon devised a way to calculate π based on probability. You can try it yourself at the Exploratorium's Pi Toss exhibit.

Download this article as a PDF.

Shown: Thomas Degeorge (1786–1854), The Death of Archimedes (detail), 1815. Collection of the Musée d’Art Roger-Quilliot Museum [MARQ], City of Clermont-Ferrand, France.

Source : www.exploratorium.edu

Pi

Pi

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This article is about the mathematical constant. For the Greek letter, see Pi (letter). For other uses, see Pi (disambiguation).

Part of a series of articles on the

mathematical constant π

3.1415926535897932384626433...

Uses

Area of a circleCircumferenceUse in other formulae

Properties

IrrationalityTranscendence

Value

Less than 22/7ApproximationsMemorization

People

ArchimedesLiu HuiZu ChongzhiAryabhataMadhavaLudolph van CeulenSeki TakakazuTakebe KenkoWilliam JonesJohn MachinWilliam ShanksSrinivasa RamanujanJohn WrenchChudnovsky brothersYasumasa Kanada

History ChronologyBook In culture LegislationPi Day Related topics

Squaring the circleBasel problemSix nines in πOther topics related to π

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The number π (/paɪ/; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159. It is defined in Euclidean geometry[a] as the ratio of a circle's circumference to its diameter, and also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.[1] It is also referred to as Archimedes's constant.[2][3]

As an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.

It is known that π is a transcendental number:[2] It is not the root of any polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.

Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later, when the Madhava–Leibniz series was discovered by the Kerala school of astronomy and mathematics, documented in the , written around 1530.[4][5]

The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits.[6][7] The primary motivation for these computations is as a test case to develop efficient algorithms to calculate numeric series, as well as the quest to break records.[8][9] The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.

Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. In more modern mathematical analysis, it is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and sciences having little to do with the geometry of circles, such as number theory and statistics, and in almost all areas of physics. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of the scientific community. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.

Contents

1 Fundamentals 1.1 Name 1.2 Definition

1.3 Irrationality and normality

1.4 Transcendence

1.5 Continued fractions

1.6 Approximate value and digits

1.7 Complex numbers and Euler's identity

2 History 2.1 Antiquity

2.2 Polygon approximation era

2.3 Infinite series

2.3.1 Rate of convergence

2.4 Irrationality and transcendence

2.5 Adoption of the symbol π

3 Modern quest for more digits

3.1 Computer era and iterative algorithms

3.2 Motives for computing π

3.3 Rapidly convergent series

3.4 Monte Carlo methods

3.5 Spigot algorithms

4 Role and characterizations in mathematics

4.1 Geometry and trigonometry

4.2 Units of angle 4.3 Eigenvalues 4.4 Inequalities

4.5 Fourier transform and Heisenberg uncertainty principle

4.6 Gaussian integrals

4.7 Projective geometry

4.8 Topology 4.9 Vector calculus

4.10 Cauchy's integral formula

4.11 The gamma function and Stirling's approximation

4.12 Number theory and Riemann zeta function

4.13 Fourier series

4.14 Modular forms and theta functions

Source : en.wikipedia.org