The Man Who Invented Pi

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