认识发明π的人:威廉·琼斯和他的圆 - Peak one Temple

在1706年,一个名叫威廉•琼斯的数学老师第一次使用了一个符号来代表圆周率π,一个用数值可以接近却永远无法达到的理想概念。

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任意圆周长与直径的恒定比值的历史和人们渴望测量的历史一样悠久,然而这个今天广为人知的比值π是起源于十八世纪早期。在这之前,这个比值用中古拉丁文晦涩地表示为:quantitas in quam cum multiflicetur diameter, provenietcircumferencia(这个量乘直径会得到周长)。

人们广泛认为是出生在瑞士的伟大数学家莱昂哈德•欧拉(1707-1783)将符号π引入普遍使用。事实上,在欧拉出生前一年的1706年,π第一次以它的现代含义出现在一个自学成才的数学老师威廉•琼斯的第二本书《新数学导论》中,这本书是基于他的教学笔记编写而成的。

在符号π出现前,像22/7和355/113的近似值被用来表示这个比值,这带来一种这个比值是个有理数的印象。尽管琼斯没有作证明,但是他相信π是个无理数,一个无限不循环小数,它不可能完全用数字形式表达。在《最新数学导论》中,他指出“…周长与直径的比值不可能由数字准确地表达”。因此需要用一个符号来表达这个可以接近却无法达到的理想概念。为此,琼斯认为只有一个纯的理想的符号才能满足需要。

在之前一个世纪,符号π被同时是教区长的数学家威廉•奥特雷德(1575-1660)用作另外的含义。在他的书《数学之钥》(在1631年第一次出版),他使用π代表给定圆的周长,所以他的π会随圆的直径的变化而变化,而不是现在代表一个常数。那时候圆的周长用'periphery'表示,因此用希腊对应字母“π”来表示。琼斯对π的使用是一个重要的奥特雷德没有实现的哲学进步,尽管奥特雷德引入了其他的数学符号,比如::表示比例以及'x'作为乘法的符号。

在奥特雷德去世的1660年,数学家约翰•科林斯(1625-1683)获得了奥特雷德数学图书馆中的一些书和论文,而琼斯也是通过约翰•科林斯获得了这些资料。
π的无理数特性直到1961年才被约翰•兰伯特(1728-1777)证明,然后在1882年费迪南•林德曼(1852-1939)证明了π是非代数的无理数,是一个超越数,即不能是任意次数的有理系数代数方程的解。有两个类型无理数的发现并没有贬低琼斯认识到周长与直径的比值不能用有理数表示的成就。

在第一次使用符号π之外,琼斯是非常令人感兴趣的,因为他与很多十八世纪的关键数学人物、科学人物与政治人物的联系。他还负责建设一个伟大的科学图书馆和数学档案馆,它们在他的赞助人麦克莱斯菲尔德家族的手中从当时一直保存了将近300年到现在。

尽管琼斯是带着数学成就去世的,但是他的出身是普通的。在大约1675年,他出生在安格尔西岛的一个小农场中。他唯一接受过的正式教育实在当地的慈善学校,在那里他展示出了数学才能,然后他被安排到伦敦的一个商人的帐房工作。后来,他航行到西印度群岛而且开始对航海感兴趣。后来他在一艘军舰上当数学老师。在1702年十月他参加了比戈战役,这场战役中英国人成功地拦截了由法国护送回西班牙西北部港口的西班牙舰队。胜利的水兵登上岸寻找金银,而根据廷茅斯男爵1807年的回忆录,对于琼斯来说最大的战利品是梦寐以求的文学珍品。

在琼斯回到英国后,他离开了海军然后开始在伦敦教数学,可能一开始在一个咖啡屋收取少量费用给人们上课。1702年,他出版了他的第一本书,《新实用航海艺术的纲要》。在这不久以后,他成为了菲利普约克的老师。后来菲利普约克(1690-1764)成为阿德威克第一任伯爵,他任大法官而且为介绍他的导师琼斯提供了无价的资源。

在大约1706年,在琼斯发表了《新数学导论》时,他第一次得到了艾萨克牛顿的关注,他在其中解释了牛顿的微积分方法和其他数学新观念。在1708年,琼斯可以获得克林斯的图书馆和档案馆的丰富资料,包括许多牛顿在17世纪60年代写的信和论文。这些提高了公众对琼斯的兴趣对他的名声很有帮助。

出生相离半个世纪,克林斯和琼斯从来没有相见,然而由于图书馆和数学档案馆历史将这两个人永久的联系在一起。图书馆和数学档案馆由克林斯建立,琼斯继续管理,在他俩对收集书籍的热情下发展壮大。克林斯是贫困牧师的儿子,他在一个图书商那里当学徒。像琼斯一样他基本上也是自学,也走向海洋学习航海。在他回到伦敦后,他靠当老师和会计谋生。他拥有几个不断获利的岗位而且擅长理顺复杂的账目。

克林斯有个普通的志向就是开一个书店,但是他没有积累足够的资金。然而在1667年,他被选入皇家学会,成为不可缺少的成员,协助学会秘书亨利•奥尔登伯格处理数学事务。从那时开始,克林斯与牛顿以及很多顶尖的英国和国外数学家一样,代表学会起草数学笔记。

在1709年当琼斯申请基督医院数学学校校长时,他带了牛顿和埃蒙德哈雷的推荐信,尽管有这些,但他还是失败了。然而,琼斯之前的学生,现正从事法律事业的飞利浦约克他的导师推荐给托马斯帕克爵士(1667-1732),他是一个成功的律师并且在下一年将要成为下一人最高法院首席法官。琼斯加入了他的家庭,并成为他儿子乔治(1697-1764)的导师。这是他与帕克家庭常年交往的开始。

在那时,琼斯买下了克林斯的图书馆和档案馆,牛顿和德国数学家莱布尼茨正在辩论是谁先发明了微积分。在克林斯的数学论文中,琼斯发现了牛顿最早使用微积分的副本《分析》(1669),他在1711年出版了这本书。这本书之前仅仅是不公开的流传。从1703年担任皇家学会会长的牛顿不情愿让他的成果发表而且小心翼翼地保护自己的知识产权。然而,他把琼斯视为他的支持者。

在1712年,琼斯加入了皇家学会建立的确认微积分的最先发明者的委员会。琼斯把克林斯的论文和牛顿关于微积分的信件提供给了委员会,并且形成了一个有关争端的报告,这个报告《Commercium Epistolicum》在那一年发表,它的大部分内容都是基于克林斯的论文和牛顿关于微积分的信件撰写。尽管这个报告是匿名的,但它被牛顿编辑,所以很难认为是公正的。不出意料,它是站在牛顿一边的。(今天,大家认为牛顿和莱布尼茨都独立地发明了微积分,尽管莱布尼茨的标记法优于牛顿的而且是目前普遍使用的。)

到1712年,琼斯已经有稳固的数学成就了。在1718年,他的赞助人托马斯帕克爵士被成为大法官并且在1721年被封为麦克莱斯菲尔德伯爵。在那时,他已经用当时总计18350英镑购买了锡伯恩地产和城堡。锡伯恩城堡同样也成为了琼斯的家,在那时他几乎已经是一个家庭成员了。除了法律帕克对许多学科包括科学和数学有学术兴趣,而且他对科学和艺术还是一个慷慨的赞助商。他作为皇家天文学家在1721年“约会”哈雷彗星过程中有很大的影响。

但是在第一伯爵的人格中也有对立面。他似乎在拥有很强的能力和抱负的同时对财富也有危险强烈的欲望。他被指控贩卖大法官职务给最高竞买人,并且允许将让投资者的资金被滥用。在1725年帕克从大法官职位辞职,但是他仍被控告。他被罚缴纳30000英镑,并且被禁足在伦敦塔6周直到罚金缴齐。他的一些资产被变卖,他被枢密院除名。但是他并没有丧失锡伯恩,锡伯恩由麦克尔斯菲尔德家族拥有到现在。在1727年,他是牛顿葬礼送葬者之一,这恢复了一些他的尊严。

托马斯的儿子乔治帕克在1722年成为了沃灵福德的一个议员,并在锡伯恩度过了大量时间,在那里在琼斯的指导下,他丰富老了琼斯带来的图书馆和档案馆。乔治帕克对天文很有兴趣,在一个天文家朋友詹姆斯布拉德利(在1742年哈雷去世时成为第三皇家天文家)的帮助下,他在锡伯恩建立了一个天文台。

到1718年,琼斯将时间花费在锡伯恩和临近伦敦红狮广场的蒂博尔德的宫殿。在许多有影响力的数学家、天文家和自然哲学家中,他结识了罗杰科茨(1682-1716),他是剑桥第一个布卢米安天文学教授,他被很多人认为是那一年代牛顿之后最有才能的英国数学家。他被委托修订牛顿原理第二版的出版物。

当牛顿和科茨关系紧张时,琼斯便作他们的中间人。他显然有影响力而且相当的机智。在一封信中科茨对琼斯写道:“有件事情我自己不能很好地处理,需要您的协调…”。这件微妙的事情是对牛顿的一个方法改良的建议牛顿有难以相处的人格,必须小心对待。而琼斯可以做得很好。牛顿原理第二版在1713年出版,得到很大的赞扬。

牛顿在大多数时期像是高耸的巨人,科学界活在他的阴影下。琼斯和天文学学家、数学家约翰梅钦有广泛的通信。约翰梅钦从1718年开始在皇家学会担任秘书近30年。他也是学会调查微积分发明的委员会成员。他在格雷沙姆学院任天文学教授近40年,研究月球运动理论并且认为他自己是这一学科的专家。在写给琼斯的一封信中,他用富于幻想语言来抱怨牛顿的月球运动理论。

她(月球)通知我说他(牛顿)在她生命的整个过程中污辱她,公布说她因不规则和各种罪恶应感到内疚,继续说没有活着的人可以在任何时间发现她的位置。

他继续写道,他梅钦,知道月亮在什么地方而且他有能力获得“Lord Treasurer”提供发现海上经度的10000英镑,因为他的月球运动理论可以提高月亮航用表的准确度。

尽管梅钦没有获得那奖金,他的月球运动理论被描述为依照重力的月球运动规律并且在牛顿死后的1729年添加到了牛顿定理的英文版中。

梅钦也在周长与半径比值方面做了一系列工作,他的计算方法快速收敛。他的计算结果被印刷在琼斯1706年的书中“超过100个地方可以验证正确;由准确、文思敏捷、真正有天才的约翰梅钦先生计算..”梅钦使用其和收敛于π的无穷级数来计算。用数学术语意味着,无论有多少项求和这个和的值与π的值总是有差距尽管差距很小。梅钦使用的无穷级数里的项正负交替,所以和的值交替地小于和大于π。

琼斯也和海外人士保持联系。其中一位特别兴趣的是住在美洲的教友派信徒学者詹姆斯洛根(1674-1751)。洛根出生于爱尔兰,被教友派领导人和宾夕法尼亚州建立者威廉佩恩邀请作他的秘书。他把那里建设得很兴旺,最终买下了斯坦顿大农场,在那里他从50多岁退休并开始追寻他的兴趣包括数学和植物学。他拥有的图书馆有超过3万本书,是美国18世纪最出名的图书馆之一并且后来赠给费城。

在1732年,洛根写信给琼斯,信中内容与一个发明相关:“这里的一个年轻人…是非常有天赋的”。这个年轻人是托马斯戈弗雷(1704-1749),他是一个装玻璃工人,在1730年10月发明了一个可以在海上准确应用的仪表,因为这个仪表有一个单向透视玻璃太阳和地平线的反射图像排成一行。任意两个天体例如月亮和一个星星可以通过移动一个包含镜子的旋转臂排成一排,而且可以从量表中读出角度。这意味着船的移动不会干扰角度测量,因为物体和图像会同时移动。这是一个精巧的仪表。洛根认为可以用它确定海上经度。这个仪表就是现在我们知道的哈德利四分仪,尽管实际上是个八分仪。英国和美国都索要了这个发明的归属。英国天文学家约翰哈德利(1682-1744)在1730年的夏天制作了一个这样的仪表而且在接下来的五月把一个报告给了皇家学会。

洛根写了一个私人信件描述戈弗雷的发明给哈雷,然后皇家学会的会长称他为“尊敬的朋友”。这是一个友好的科学的沟通,而皇家学会照例没有阅读这个信件。洛根向琼斯询问这一遗漏。琼斯后来在1734年一月和学会提出这个议题,戈弗雷作为仪表的发明者的地位被确立,尽管不是第一发明者。

在过了一些年的1736年琼斯写信给洛根,为没有及时回复道歉,他写道:

我的事务需要我全神贯注而且占据了我的思想以至于我有很少或者几乎没有时间考虑其他的事情甚至是数学。过去的这18年我缺少想法,我现在那些改进几乎是一个陌生的人。

但是在那个时间过后琼斯有关于数学学科的通信。可能是他不想鼓励洛根给他一些其他的发现。洛根是一个不知疲倦的通信者,他写的信比琼斯回复的信多很多。

当然琼斯脑海里是有其他东西的。像许多其他的研究科学的人,琼斯对经度问题感兴趣。他给皇家学会写信有关于当温度变化时时钟保持精确时间的课题。

他担任学会委员会成员并且在1749 年成为他的副会长。他的收入因工作清闲但报酬优厚的职位而大涨,这个职位是由他之前的学生建立的。他在阿德威克的影响下担任和平秘书,在乔治帕克的帮助下担任财政部副出纳员。然而他仍然在那时候经常发生的银行破产的作用下经历多次经济危机

琼斯在1731年完成了第二次婚姻,娶了比他小30岁的玛丽尼克斯,他们有三个孩子。在1747年他被选为育婴医院管理者,这时乔治帕克是副院长。是乔治让贺加斯为琼斯作画。尽管琼斯在这幅画中看起来令人注目,但是他被报道是一个矮小脸不长的威尔士人并且经常用粗暴和自由对待他的数学朋友。尽管如此,就像我们已经看到的,他知道在必要时如何变得机智而且展示盛意。

在他1749年74岁去世之后。皇家学会职员和图书馆管理员约翰罗伯特森说他去世时的情况比很多数学家好。他唯一存活的儿子,也叫威廉,那时只有三岁。他为人知的名字是奥连塔尔琼斯,他是一个出色语言学家和文献学者而且他精通印度法律而且他被正式封爵。

在1750年,乔治帕克撰写了一篇论文,这篇论文被皇家学会阅读而且被命名为评论太阳和月亮年。乔治是采用阳历最重要的支持者而且在1752年将新年从3月25日改到1月1日。有些人可能认为日历的修订是威廉琼斯科学遗产的一部分。在同一年,帕克被选为皇家学会会长,他直到去世都担任这一职务。

按照琼斯的意愿,他把学术书籍给乔治帕克作为他接受了帕克很多帮助的证明与鸣谢。帕克从琼斯继承的科学书籍和档案馆里的论文保存在锡伯恩的图书馆中。得到这些资料受到了严格的控制,尽管需要承认的是他们代表了他们在私人手中的最重要的书籍。在2000年剑桥大学图书馆在遗产彩票基金一笔基金的帮助下花费6370000英镑购买了档案馆的书信和论文。在2005年麦克莱斯菲尔德图书馆最终在索斯比以世界第六大销售额卖掉。

在琼斯的一生中,他将赞助商留住的能力十分重要而且他为他们服务得很好。从历史的角度来看,琼斯为麦克莱斯菲尔德做出贡献远大于他从赞助商的获取,正是这样,他为世界留下了智力遗产。

本文参考《The Man Who Invented Pi》,作者帕特里夏·罗斯曼(Patricia Rothman)是伦敦大学学院数学系的荣誉研究员。以下引述全文。

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

williamjones.jpg
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.

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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