罗杰·柯特斯Roger Cotes

罗杰·柯特斯的母亲是Grace Farmer，来自莱斯特郡的Barwell，他的父亲是Robert Cotes，是Burbage的教区长。柯特斯有一个比他大一岁的兄弟Anthony，和一个比他小一岁的姐妹Susanna。他就读于莱斯特学校，到十二岁时，他的老师已经意识到他具有非凡的数学天赋。他的叔叔，牧师John Smith，热衷于给柯特斯一切机会发展这些才能，因此柯特斯去和他住在一起，以便能亲自接受辅导。柯特斯后来就读于伦敦著名的圣保罗学校，但他继续接受叔叔的指导，在柯特斯在伦敦上学期间，两人就数学主题交换了信件。

柯特斯于1699年4月6日作为自费生注册入剑桥大学三一学院，这意味着他没有奖学金，需自付学院食宿费用。他于1702年获得学士学位，并留在剑桥，于1705年被选为研究员。1706年1月，他被提名为首位普卢米安天文学与实验哲学讲席教授。对于当时年仅23岁的柯特斯来说，这是一项非凡的成就。然而，他的非凡才能已得到剑桥许多人的充分赏识，例如威廉·惠斯顿，他很快与之建立了友谊。艾萨克·牛顿和威廉·惠斯顿都推荐柯特斯担任该讲席，三一学院院长理查德·本特利也是如此。然而，也有人反对他的任命，其中最引人注目的是皇家天文学家约翰·佛兰斯蒂德。到柯特斯于1707年10月16日正式当选为普卢米安教授时，他已于前一年当选为更负盛名的研究员，并获得硕士学位。梅利在[2]中给出了该讲席设立的背景：-

柯特斯是罗切斯特副主教托马斯·普卢姆（1630-1704）设立的剑桥讲席的首位担任者，普卢姆遗赠了近2000英镑用于维持一位教授并建造一座天文台。在普卢姆遗赠之前，本特利已经起草了三一学院天文台的计划。天文台最终设于三一学院国王门或大门之上，连同普卢米安教授的住所。

尚不完全清楚柯特斯作为观测天文学家的角色有多成功。首先，关于剑桥天文台仪器质量的记载有些矛盾。柯特斯设计了一台中星仪，以补充已购买或捐赠的仪器收藏。例如，艾萨克·牛顿捐赠了一座至今仍存于三一学院的钟。我们上文提到的三一学院院长本特利声称天文台拥有“欧洲最好的仪器”，但在那里工作的一位助手写信给约翰·佛兰斯蒂德说：“我在那里没有看到任何值得您注意的东西。”真相可能介于两者之间，因为三一学院院长自然会夸耀设施，而只在那里工作了很短时间的助手可能是在试图取悦约翰·佛兰斯蒂德。就柯特斯所做的观测而言，也许最重要的是1715年4月22日的日全食。然而，爱德蒙·哈雷在Philosophical Transactions of the Royal Society的一篇论文中描述了这一事件，他说柯特斯：-

……不幸被过多的陪伴所困扰，因此尽管天空非常有利，他还是错过了日食开始的时间和全食的时间。

柯特斯本人就这次日食写信给艾萨克·牛顿，在信中他解释说他的助手发现了一种确定日食中点的方法，而他[3]：-

……对我喊道：“现在是中间了”，尽管我当时并不知道他是什么意思。

这一切都说明不了柯特斯作为观测者的献身精神有多高，但他确实记录了一些关于这次日食和其他天文事件的重要事实。然而，他的数学才能使他在英格兰同代人中仅次于艾萨克·牛顿。在考察他的数学贡献之前，让我们注意到他于1711年11月30日当选为皇家学会会士，1713年3月30日被任命为执事，1713年5月31日被任命为牧师。

从1709年到1713年，柯特斯的大部分时间都用于编辑艾萨克·牛顿的Principia第二版。他不仅仅是校对这部著作，而是认真地研究它，温和而坚持不懈地与艾萨克·牛顿争论各种问题。例如，在[6]中讨论了一个1711年发生在柯特斯和艾萨克·牛顿之间关于水从圆柱形容器孔中流出的速度的讨论。在讨论中，他们给出了2的四次方根的各种近似值，其值约为1.189207115。艾萨克·牛顿给出了以下有理的近似值（我们添加小数值以查看其精度）

$\large\frac{6}{5}\normalsize = 1.200000000$
$\large\frac{13}{11}\normalsize = 1.181818182$
$\large\frac{25}{21}\normalsize = 1.190476190$

而柯特斯给出了

$\large\frac{44}{37}\normalsize = 1.189189189.$

在两人通信之初，语气非常友好。然而，在任务接近尾声时，有迹象表明他们对彼此冷淡了（有关这些信件的细节，见[3]）。特别是，尽管艾萨克·牛顿在他为这一版所写序言的第一稿中感谢了柯特斯，但他在最终出版时删去了这些感谢。柯特斯自己写了一篇有趣的序言，其中他解释了自然哲学研究是如何发展的。首先，柯特斯解释说，是亚里士多德的方法，涉及命名隐藏的性质。然后，根据柯特斯的说法，出现了所有物质都是同质的观念。他认为这些方法是改进，但仍保留了亚里士多德方法的某些弱点。尽管他在这里没有具体点名勒内·笛卡儿和哥特弗里德·威廉·莱布尼茨，但这显然是对他们思想的攻击。最后，柯特斯说，是基于先进行实验而不带先入之见，然后从结果中推断世界如何运作的方法。这些是艾萨克·牛顿的方法，导致了自然界基本力如何运作的确立。

柯特斯一生只发表了一篇论文，即Logometria，发表于1714年3月的Philosophical Transactions of the Royal Society上，他将此文献给了爱德蒙·哈雷。文中包含（用柯特斯在给艾萨克·牛顿的信[3]中自己的话来说）：-

……几何学中一种新的作图法，在我看来非常容易、简单且普遍。

在这篇论文中，柯特斯给出了一种通过连分数的渐近分数来求有理逼近的方法，而[6]的作者认为，这解释了他如何得到我们上文提到的2的四次方根逼近值$\large\frac{44}{37}\normalsize$。

柯特斯对他关于对数曲线的求长特别满意，正如他在1712年给朋友威廉·琼斯的信中所表明的那样。特别是他对对数的工作引导他研究了曲线$r = \Large\frac{a}{\theta}\normalsize$，他将其命名为倒数螺线。柯特斯扩展了皮埃尔·伐里农的工作，他对阿基米德螺线和阿波罗尼奥斯的抛物线进行了求长，这个问题最初由皮埃尔·德·费马提出，他证明两者具有相同的积分。他在这里的工作基于公式

$\ln(\cos q + i \sin q) = i q$。

威廉·琼斯敦促柯特斯在Philosophical Transactions of the Royal Society上发表他的工作，但柯特斯拒绝了，他希望支持剑桥并与剑桥大学出版社一起出版。他的早逝使得这些工作未能在其生前发表。

柯特斯发现了关于$n$次单位根的一个重要定理，给出了$e$的连分数展开，发明了角的弧度制，预见了最小二乘法，发表了正切和正割的图形，并发现了一种对具有二项式分母的有理分式进行积分的方法。他在对数理论、积分学、数值方法（特别是插值以及十八类代数函数的积分表构造）方面的实质性进展，使艾萨克·牛顿说道：-

……如果他活得更久，我们或许本可以知道些什么。

根据Edleston3，柯特斯死于一种：——

……伴有剧烈腹泻和持续谵妄的发热。

四天后，他被安葬在三一学院的小教堂里。

柯特斯希望与剑桥大学出版社一起出版的一些工作最终由托马斯·辛普森在The Doctrine and Application of Fluxions（2卷，伦敦，1750年）中出版。Robert Smith编辑了柯特斯的主要遗作，即Harmonia mensurarum，于1722年出版。在此解释Robert Smith是谁以及他如何与柯特斯互动是合适的。事实上，他是柯特斯的叔叔，即牧师John Smith的儿子，并且当他的叔叔让他小时候住在自己家里时，他就与柯特斯成为了朋友。后来Robert Smith在柯特斯担任Plumian教授时是他的助手，并最终接替他成为Plumian教授。正是Smith，在柯特斯去世多年后，当他担任三一学院院长时，让人竖立了柯特斯的半身像。这座半身像，如上所示，现在在克里斯托弗·雷恩图书馆中。

让我们回到柯特斯的遗著Harmonia mensurarumⓉ（《测度之和，或分析与综合的记述，并推进角度测度，以及其他数学著作》）。除了重印Logometria之外，它还包含三部数学著作：

1. Aestimatio errorum in mixta mathesis。Ⓣ（数学各领域中的误差评估）
2. De methodo differentiali Newtoniana。Ⓣ（艾萨克·牛顿的微分方法）
3. Canonotechnia。Ⓣ（技术规则）

第一部分涉及平面三角形和球面三角形，曾被天文学家广泛使用。其中包含了对误差理论的早期研究。第二部分发展了艾萨克·牛顿的插值方法，在研究彗星轨道时特别有用。第三部著作研究数值积分，并且还包含了对插值法的进一步贡献。

1738年，即柯特斯去世22年后，Smith出版了柯特斯关于实验物理学的讲座Hydrostatical and pneumatical lectures。

Roger Cotes' mother was Grace Farmer, who came from Barwell in Leicestershire, and his father was Robert Cotes who was the rector of Burbage. Roger had a brother Anthony one year older than himself, and a sister Susanna who was one year younger. He attended Leicester School and by the age of twelve his teachers had already realised that he had an exceptional mathematical talent. His uncle, the Reverend John Smith, was keen to give Roger every chance to develop these talents and so Roger went to live with him so that he might be personally tutored. Roger later attended the famous St Paul's School in London, but he continued to be advised by his uncle and the two exchanged letters on mathematical topics during the time that Roger spent at school in London.

Roger matriculated at Trinity College, Cambridge, on 6 April 1699 as a pensioner, meaning that he did not have a scholarship and paid for his own keep in College. He graduated with a B.A. in 1702 and remained at Cambridge where he was elected to a fellowship in 1705. In January 1706 he was nominated to be the first Plumian Professor of Astronomy and Experimental Philosophy. This was a remarkable achievement for Cotes who, at that time, was on 23 years of age. His exceptional abilities had been fully appreciated, however, by many at Cambridge such as William Whiston with whom he had quickly formed a friendship. Both Newton and Whiston recommended Cotes for the Chair, as did Richard Bentley who was master of Trinity College. There were some, however, who opposed his appointment, the most high profile of whom was Flamsteed, the astronomer royal. By the time that Cotes was formally elected as Plumian Professor on 16 October 1707 he had, in the previous year, been elected to a more prestigious fellowship as well as being awarded his M.A. Meli gives the background to the establishment of the chair in [2]:-

Cotes was the first occupant of the Cambridge chair established by Thomas Plume (1630 - 1704), archdeacon of Rochester, who bequeathed nearly £2000 to maintain a professor and erect an astronomical observatory. Plans for an observatory at Trinity had already been drafted by Bentley before Plume's bequest. The observatory was eventually housed over the king's or great gate at Trinity College, together with living quarters for the Plumian professor.

It is not entirely clear how successful Cotes was in his role as an observational astronomer. In the first place there are somewhat contradictory accounts of the quality of the instruments in the Cambridge observatory. Cotes designed a transit telescope to add to a collection of instruments which had been purchased or donated. For example Newton donated a clock which still survives at Trinity College. Bentley, the master of Trinity College we mentioned above, claimed that the Observatory had "the best instruments in Europe" but an assistant who worked there wrote to Flamsteed saying "I saw nothing there that might deserve your notice". The truth is probably somewhere in between, since it would be natural for the master of Trinity to boast of the facilities, while the assistant, who only worked there for a short time, was probably trying to please Flamsteed. In terms of the observations that Cotes made, perhaps the most significant was the total eclipse on 22 April 1715. However, Halley describes this event in a paper in the Philosophical Transactions of the Royal Society where he says that Cotes:-

... had the misfortune to be opprest by too much company, so that though the heavens were very favourable, yet he missed both the times of the beginning of the eclipse and that of total darkness.

Cotes himself wrote a letter to Newton concerning the eclipse in which he explained that his assistant had discovered a method to determine the mid-point of the eclipse and he [3]:-

... called out to me, "Now's the middle", though I knew not at that time what he meant.

None of this speaks very highly of Cotes' dedication as an observer, but nevertheless he did note some important facts concerning this eclipse and other astronomical events. However, his mathematical abilities put him second only to Newton from his generation in England. Before going on to look at his mathematical contributions let us note that he was elected a fellow of the Royal Society on 30 November 1711, was ordained a deacon on 30 March 1713, and was ordained a priest on 31 May 1713.

From 1709 until 1713 much of Cotes' time was taken up editing the second edition of Newton's Principia. He did not simply proof-read the work, rather he conscientiously studied it, gently but persistently arguing points with Newton. For example in [6] a discussion is considered which took place between Cotes and Newton in 1711 concerning the velocity of water flowing from a hole in a cylindrical vessel. During the discussion they gave various approximations to the fourth root of 2 which is approximately 1.189207115. Newton gave the following rational approximations (we add decimal values to see their accuracy)

$\large\frac{6}{5}\normalsize = 1.200000000$
$\large\frac{13}{11}\normalsize = 1.181818182$
$\large\frac{25}{21}\normalsize = 1.190476190$

while Cotes gave

$\large\frac{44}{37}\normalsize = 1.189189189.$

At the beginning of the correspondence between the two men the tone is very friendly. However, toward the end of the task there are signs that they are cooling towards one another (see [3] for details of these letters). In particular although Newton thanked Cotes in the first draft of a preface he wrote to this edition, he deleted these thanks for the final publication. Cotes himself wrote an interesting preface of his own in which he explained how the study of natural philosophy had developed. First, Cotes explained, came Aristotle's method which involved naming hidden properties. Then, according to Cotes, came the ideas that all matter was homogeneous. He saw these methods as improvements, yet still retaining certain of the weaknesses of Aristotle's approach. Although he does not specifically name Descartes and Leibniz here, it is clearly an attack on their ideas. Finally says Cotes, comes the method based on first conducting experiments without having preconceived ideas, and then deducing how the world works from the results. These were the methods of Newton which led to establishing how the basic forces of nature operated.

Cotes only published one paper in his lifetime, namely Logometria, published in the Philosophical Transactions of the Royal Society for March 1714, which he dedicated to Halley. It contains (in the words that Cotes used himself in a letter to Newton [3]):-

... a new sort of construction in geometry which appear to me very easy, simple and general.

In this Cotes gave a method of finding rational approximations as convergents of continued fractions, and the author of [6] suggests that this explains how he found the approximation $\large\frac{44}{37}\normalsize$ to the fourth root of 2 which we mentioned above.

Cotes was particularly pleased with his rectification of the logarithmic curve as he made clear in a letter to his friend William Jones in 1712. In particular his work on logarithms led him to study the curve $r = \Large\frac{a}{\theta}\normalsize$ which he named the reciprocal spiral. Cotes extended the work of Varignon when he rectified the Archimedean spiral and the parabola of Apollonius, a problem first proposed by Fermat, showing that both have the same integral. His work here was based on the formula

$\ln(\cos q + i \sin q) = i q$.

Jones urged Cotes to publish his work in the Philosophical Transactions of the Royal Society, but Cotes resisted this, wishing to support Cambridge and publish with Cambridge University Press. His early death was to prevent publication in his lifetime.

Cotes discovered an important theorem on the $n$th roots of unity, gave the continued fraction expansion of $e$, invented radian measure of angles, anticipated the method of least squares, published graphs of tangents and secants, and discovered a method of integrating rational fractions with binomial denominators. His substantial advances in the theory of logarithms, the integral calculus, in numerical methods particularly interpolation and table construction of integrals for eighteen classes of algebraic functions led Newton to say:-

... if he had lived we might have known something.

According to Edleston [3], Cotes died of a:-

... fever attended with a violent diarrhoea and constant delirium.

He was buried four days later in the chapel of Trinity College.

Some of the work which Cotes hoped to publish with Cambridge University Press was published eventually by Thomas Simpson in The Doctrine and Application of Fluxions (2 Vols, London, 1750). Robert Smith edited Cotes' major posthumous work, the Harmonia mensurarum which appeared in 1722. It is fitting at this point to explain who Robert Smith was, and how he interacted with Cotes. In fact he was the son of Cotes' uncle, the Reverend John Smith, and had become friends with Cotes when his uncle had him live in his house as a boy. Later Robert Smith was Cotes' assistant when he was Plumian Professor, and eventually succeeded him as Plumian professor. It was Smith who, many years after Cotes' death, when he was master of Trinity College, had a bust of Cotes erected. This bust, which is shown above, is now in the Wren library.

Let us return to Cotes' posthumous work the Harmonia mensurarum Ⓣ. As well as reprinting Logometria it contains the three mathematical works:

1. Aestimatio errorum in mixta mathesis. Ⓣ
2. De methodo differentiali Newtoniana. Ⓣ
3. Canonotechnia. Ⓣ

The first concerns plane and spherical triangles and was much used by astronomers. It contains an early study of the theory of errors. The second develops Newton's methods of interpolation and was particularly useful in studying orbits of comets. The third work studies numerical integration and also includes further contributions to interpolation.

In 1738, 22 years after Cotes died, Smith published the lectures which Cotes had given on experimental physics Hydrostatical and pneumatical lectures.