托马斯·辛普森Thomas Simpson

托马斯·辛普森的父亲是一名织工。辛普森几乎没有接受过正规教育。他确实曾在马基特博斯沃思上过一段时间学，但他的第一份工作是织工。他自学了数学，这在当时的织工中并不罕见，我们将在下面看到。他离开家乡，到沃里克郡的纽尼顿担任校长。大约从1725年，即辛普森十五岁时，到1733年左右，他在纽尼顿教授数学。

辛普森在Nuneaton寄宿在一位名叫Swinfield的女士家中，并于1730年与她结婚。他们有一个女儿Elizabeth生于1736年，一个儿子辛普森生于1738年。事实上，辛普森和他的妻子在孩子们出生前就离开了Nuneaton。他的传记作者所报告的原因如下。他[2]：-

……1733年不得不逃往Derby，因为他或他的助手在一次占星活动中装扮成魔鬼吓坏了一个女孩。

这肯定与他被称为以下名字的事实相符：-

……纽尼顿、博斯沃思及周边地区的预言家。

他在德比究竟待了多久不得而知，但我们确知从1736年起他与家人住在伦敦。他是斯皮塔菲尔德数学会的早期成员，1736年49名成员之一。该学会运作得像一个工人俱乐部，我们知道对于一个教数学的织工来说，这是自然而然的选择，因为到1744年其成员中：——

……约半数是织工，其余通常是酿酒师、铜匠、面包师、砌砖匠。

辛普森是在伦敦咖啡馆授课的巡回讲师群体中最杰出的一位。这或许显得奇怪，但实际上当时咖啡馆有时被称为便士大学，因为它们提供廉价的教育。它们收取一便士的入场费，然后顾客一边喝咖啡一边可以听讲座。不同的咖啡馆迎合特定的兴趣，如艺术、商业、法律和数学。例如亚伯拉罕·棣莫弗这些年以圣马丁桑德斯·麦克兰恩的斯劳特咖啡馆为基地，而辛普森的朋友威廉·琼斯则靠在圣保罗教堂墓地的柴尔德咖啡馆等咖啡馆讲课谋生。

1743年，辛普森被任命为伍利奇皇家军事学院的数学负责人。事实上，这所学院在辛普森就职前两年才成立，他在那里的任命对他所研究的数学课题产生了影响。特别是，他开始研究工程问题和与防御工事有关的问题。就职两年后，辛普森被选为皇家学会会士。在我们描述他所获得的荣誉时，我们应该注意到，他还在1758年被选为瑞典皇家科学院会士。

从1737年起，辛普森开始撰写数学教材，当年出版了A New Treatise of Fluxions[2]：——

这是一本高质量教科书，专论流数术，即无穷小微积分的牛顿版本。该主题很前沿——在1730年代写这样一本书绝非易事，当时欧洲只有少数数学家掌握了微积分。

辛普森最为人所知的是他在插值与数值积分方法方面的工作。然而，今天被称为“辛普森法则”的数值方法，尽管确实出现在他的著作中，却是他从艾萨克·牛顿那里学来的，辛普森本人也承认这一点。不过作为补偿，用于求解方程$f (x) = 0$的艾萨克·牛顿-约瑟夫·拉弗森方法，就其目前的形式而言，应归功于辛普森。艾萨克·牛顿描述了一种求解多项式方程的代数过程，约瑟夫·拉弗森后来对其作了改进。这种近似求根的方法没有使用微分学。现代迭代形式$x_{n+1} = x_n - \Large \frac {f(x_n)}{f^\prime (x_n)}$应归功于辛普森，他于1740年发表了它。

他还研究概率论，并于1740年出版了The Nature and Laws of Chance。辛普森在这一领域的许多工作是基于亚伯拉罕·棣莫弗的早期工作。事实上，他因概率和年金问题的优先权问题与亚伯拉罕·棣莫弗发生了争执。他研究误差理论，旨在证明算术平均值优于单次观测。他对这一点的论证出现在他1757年的论文An attempt to show the advantage arising by taking the mean of a number of observations in practical astronomy中。

辛普森于1743年出版了Mathematical Dissertations，讨论了椭圆绕其一条轴旋转所得固体的吸引力。他1750年的两卷本著作The Doctrine and Application of Fluxions包含了罗杰·柯特斯的工作，被许多人认为是18世纪出版的关于艾萨克·牛顿版本微积分的最佳著作。辛普森在Miscellaneous Tracts（1757年）中讨论了天文学中的问题，如岁差。

1754年，他成为Ladies Diary的编辑。从1736年来到伦敦时起，他就一直在Ladies Diary上发表文章。他回答了该出版物中提出的问题，但使用了各种笔名，如马默杜克·霍奇森、赫尔洛思伦博、Kubernetes、帕特里克·奥卡瓦纳和安东尼·沙洛。正是他在这些解答中表现出的明显数学才能，首先引起了当时其他数学家的注意。他还投稿的其他期刊有Gentleman's Magazine, Miscellanea Curiosa Mathematica和Gentleman's Diary。

在[9]中，Stigler描述了辛普森生命接近尾声时发生的一件事：-

一份新发现的手稿残片表明，辛普森与鲁杰尔·约瑟普·博斯科维奇于1760年见过面，并且鲁杰尔·约瑟普·博斯科维奇向辛普森提出了一个最小偏差回归问题。……辛普森尝试给出解析解的过程得到了解读。

查尔斯·赫顿对辛普森的以下描述（作于辛普森去世35年后）很有意思[5]：-

据说辛普森先生常与下九流为伍，和他们一起痛饮黑啤酒和杜松子酒：但必须指出，是他家人的不端行为使他无法与绅士们交往，也无法弄到更好的酒。

应当公平地指出，其他人将辛普森的品行描述为无可指责。

Thomas Simpson's father was a weaver. Thomas received little formal education. He did attend school in Market Bosworth for a while but his first job was as a weaver. He taught himself mathematics, something which was not uncommon for weavers at that time as we shall see below. He moved away from his home town to take up a position as a schoolmaster in Nuneaton, Warwickshire. From about 1725, when Simpson was fifteen years old, until around 1733, he taught mathematics in Nuneaton.

Simpson had lodgings in Nuneaton with a lady by the name of Swinfield whom he married in 1730. They had one daughter Elizabeth born in 1736 and one son Thomas born in 1738. In fact Simpson and his wife had left Nuneaton before his children were born. The reason which has been reported by his biographers is as follows. He [2]:-

... had to flee to Derby in 1733 after he or his assistant had frightened a girl by dressing up as a devil during an astrology session.

This would certainly fit in with the fact that he was known as the:-

... oracle of Nuneaton, Bosworth and the environs.

Exactly how long he remained in Derby is unknown but we do know that from 1736 he was living in London with his family. He was an early member of the Spitalfields Mathematical Society being one of 49 members in 1736. This Society operated as a working men's club and we know that it was a natural choice for a weaver who taught mathematics since of the members by 1744:-

... about half were weavers, and the rest were typically brewers, braziers, bakers, bricklayers.

Simpson was the most distinguished of a group of itinerant lecturers who taught in the London coffee houses. This may seems strange but in fact at this time coffee houses were sometimes called the Penny Universities because of the cheap education they provided. They would charge an entrance fee of one penny and then while customers drank coffee they could listen to lectures. Different coffee houses catered to specific interests such as art, business, law and mathematics. For example De Moivre used Slaughter's Coffee House in St Martin's Lane as a base during these years, and William Jones, who was a friend of Simpson, was able to make a living lecturing in coffee houses such as Child's Coffee House in St Paul's Churchyard.

In 1743 Simpson was appointed as the head of mathematics at the Royal Military Academy at Woolwich. In fact this Academy was founded only two years before Simpson took up the post and his appointment there had an impact on the mathematical topics he investigated. In particular he began research on engineering problems and problems relating to fortifications. Two years after his appointment, Simpson was elected a fellow of the Royal Society. While we are describing the honours which he received, we should note that he was also elected a fellow of the Royal Swedish Academy of Sciences in 1758.

From 1737 Simpson began to write texts on mathematics, publishing A New Treatise of Fluxions in that year [2]:-

This was a high-quality textbook devoted to the calculus of fluxions, the Newtonian version of the infinitesimal calculus. The topic was advanced -- it was no trivial exercise to write such a book in the 1730s, when the calculus was mastered by only a few mathematicians in Europe.

Simpson is best remembered for his work on interpolation and numerical methods of integration. However the numerical method known today as "Simpson's rule", although it did appear in his work, was something he learned from Newton as Simpson himself acknowledged. By way of compensation, however, the Newton-Raphson method for solving the equation $f (x) = 0$ is, in its present form, due to Simpson. Newton described an algebraic process for solving polynomial equations which Raphson later improved. The method of approximating the roots did not use the differential calculus. The modern iterative form $x_{n+1} = x_n - \Large \frac {f(x_n)}{f^\prime (x_n)}$ is due to Simpson, who published it in 1740.

He also worked on probability theory and in 1740 published The Nature and Laws of Chance. Much of Simpson's work in this area was based on earlier work of De Moivre. In fact he was involved in a dispute with De Moivre over issues of priority on the topic of probability and annuities. He worked on the Theory of Errors and aimed to prove that the arithmetic mean was better than a single observation. His justification of this appeared in his 1757 memoir An attempt to show the advantage arising by taking the mean of a number of observations in practical astronomy.

Simpson published Mathematical Dissertations in 1743 which discussed the attraction of the solid obtained by rotating an ellipse around one of its axes. His two volume work The Doctrine and Application of Fluxions in 1750 contains work of Cotes and is considered by many to be the best work on Newton's version of the calculus published in the 18th century. Problems in astronomy such as the precession of the equinoxes were discussed by Simpson in Miscellaneous Tracts (1757).

In 1754 he became editor of the Ladies Diary. He had published in the Ladies Diary from the time he came to London in 1736. He answered problems posed in this publication, but used a variety of pseudonyms such as Marmaduke Hodgson, Hurlothrumbo, Kubernetes, Patrick O'Cavannah, and Anthony Shallow. It was his obvious mathematical skills demonstrated in these solutions which first brought his to the attention of other mathematicians of the day. Other periodicals which he published in were the Gentleman's Magazine, Miscellanea Curiosa Mathematica, and the Gentleman's Diary.

In [9] Stigler describes an event which occurred near the end of Simpson's life:-

A newly discovered manuscript fragment shows that Thomas Simpson and Roger Boscovich met in 1760, and that Boscovich posed a problem in least deviations regression to Simpson. ... Simpson's attempt at an analytical solution is interpreted.

The following description of Simpson by Charles Hutton (made 35 years after Simpson's death) is interesting [5]:-

It has been said that Mr Simpson frequented low company, with whom he used to guzzle porter and gin: but it must be observed that the misconduct of his family put it out of his power to keep the company of gentlemen, as well as to procure better liquor.

It would be fair to note that others described Simpson's conduct as irreproachable.