爱德华·华林Edward Waring

爱德华·华林的父亲John Waring是个农民。他的家族有几代人住在什罗普郡的米顿。John Waring与伊丽莎白结婚，他们的儿子华林在什鲁斯伯里学校接受教育。

华林于1753年3月24日进入剑桥大学莫德林学院。他获得了米尔布里奇奖学金，并被录取为减费生（Sizar），这意味着他支付减免的学费，但基本上作为仆人来弥补学费的减免。他立即以数学能力给老师留下深刻印象，并于1757年以高级数学荣誉学位考试一等及格者（Wrangler）的身份获得学士学位。

1758年4月24日，华林被选为莫德林学院的研究员。华林最著名的著作是Miscellanea analytica de aequationibus algebraicis et curvarum proprietatibusⓉ（关于代数方程和曲线性质的分析杂录），他在接下来的几年里致力于此。他将这部著作的第一章提交给皇家学会，但此后两年没有任何进展。当华林于1759年被提名为剑桥大学卢卡斯数学讲席时，这部著作以Miscellanea AnalyticaⓉ（关于代数方程和曲线性质的分析杂录）的形式分发，以证明尽管他年轻，但有资格担任该职位。

剑桥大学圣弗瑞兹·约翰学院的威廉·鲍威尔对于谁应该填补卢卡斯数学讲席有自己的想法，并试图阻止华林被任命。他发表了一本名为Observations的小册子，批评华林并怀疑他的数学能力。华林于1760年1月25日以A reply to the observations小册子回应了这一批评。鲍威尔仍然按照自己的议程，不会轻易放弃，并立即以Defence of the observations回应。约翰·威尔逊现在写了A letter来支持华林，这足以让他在1760年1月28日被确认为卢卡斯教授，年仅23岁。

当Miscellanea AnalyticaⓉ（关于代数方程和曲线性质的分析杂录）于1762年作为完整著作出版时，华林选择称其为第二版。相当奇怪的是，尽管是第二版，它却被赋予了一个新标题Meditationes AlgebraicaeⓉ（代数沉思录）。我们将在下面进一步评论这部重要著作，涵盖方程理论、数论和几何学等主题。1763年6月2日，华林被选为皇家学会的会士。

1764年，拉朗德发表了Life of Condorcet。在这部著作中，声称英国没有一流分析学家。华林迅速对此评论做出回应，写信给皇家天文学家内维尔·马斯基林。在信中，华林指出让·勒朗·达朗贝尔、莱昂哈德·欧拉和约瑟夫·拉格朗日都曾赞扬他1762年的工作。华林写道，在书中他给出了：-

... 大约三到四百个各种类型的新命题，远多于任何其他英国作家所给出的。

华林知道他的工作没有被广泛阅读，因为他补充说，他：-

... 从未听说英格兰剑桥以外的任何读者，费心去阅读和理解它 ...

很少有人读过这本书的原因，部分是因为主题困难，部分是因为华林不善交流，部分是因为他没有良好的代数符号。将华林与保罗·鲁菲尼进行比较是合理的，后者在大约150年后，因其代数工作而遭受了同样的命运，原因也大致相同。

人们不会想到卢卡斯数学教授会去攻读医学学位，但这正是华林所做的，他于1767年获得医学博士学位。他在伦敦多家医院短暂行医，随后在剑桥的阿登布鲁克医院，最后在亨廷顿郡圣艾夫斯的一家医院。然而，到1770年他已放弃行医[8]：-

……他非常近视，举止非常害羞，因此很快就放弃了他的职业。

有人可能会问，华林如何能同时行医并担任卢卡斯讲席。嗯，他从未将授课作为其职责的一部分。有些人声称，这是因为他的思想如此深刻，以至于无法在讲座中传达，但说实话，更可能的原因是他不善沟通，笔迹几乎无法辨认。[我同意，这类问题并没有阻止其他人讲课！]

1776年，华林与Mary Oswell结婚。他们曾在什鲁斯伯里住了一段时间，但玛丽不喜欢这个城镇，于是夫妇俩搬到了华林位于庞特斯伯里普莱利的庄园。

正如我们已经注意到的，华林1762年的Miscellanea Analytica ...Ⓣ（关于代数方程和曲线性质的分析杂录）为后续著作奠定了基础。涵盖几何学的Proprietates algebraicarum curvarumⓉ（代数曲线的性质）于1772年出版。涵盖方程理论和数论的Meditationes AlgebraicaeⓉ（代数沉思）于1770年问世，1782年出版了扩充版。另一部著作Meditationes analyticaeⓉ（分析沉思）于1776年出版，1785年出版了新的扩充版。

在Meditationes AlgebraicaeⓉ（代数沉思）中，华林证明了方程根的所有有理对称函数都可以表示为系数的有理函数。他推导出一种表示对称多项式的方法，并研究了分圆方程$x^{n} - 1 = 0$。这项工作使华林成为伽罗瓦理论最早的贡献者之一。特别是，在讨论第3章问题22时，Weeks写道[3]：-

华林处理这个例子的最重要方面是四次方程的根与其预解三次方程之间的对称关系。这本质上是对称函数理论中的第一个结果（超出了第1章中出现的基本构建块），该理论的系统发展直到19世纪才出现（约瑟夫·拉格朗日、卡尔·弗里德里希·高斯等），并最终被置换群理论（埃瓦里斯特·伽罗瓦、卡米耶·若尔当……）所继承。

Meditationes AlgebraicaeⓉ（代数沉思）的第4章包含如下结果：-

k 个未知数的 k 个方程可以化为一个未知数的一个方程。

他的结果——原方程次数的乘积等于化简后单个方程的次数——被称为艾蒂安·贝祖的推广定理。

本书的其余部分涉及数论，华林在这一领域取得了一些有趣的进展。他指出，任何偶数都可以写成两个素数之和，每个奇数要么是素数，要么是三个素数之和。这一结果现在被称为Goldbach conjecture，是数学中最著名的未解决问题之一。尽管克里斯蒂安·哥德巴赫在给莱昂哈德·欧拉的信中提出他的问题远早于华林出版Meditationes AlgebraicaeⓉ（代数沉思），但仍然值得注意的是，华林的版本是第一个被发表的。

华林还陈述了如今被称为“华林定理”的内容，但没有给出证明：-

每个整数都等于不超过 9 个立方数之和。同样，每个整数都是不超过 19 个四次方数之和，依此类推……

1909 年，大卫·希尔伯特证明了：给定任意整数$n$，存在一个整数$m$（依赖于$n$），使得每个整数都是$m$个$n$次方数之和。有理由认为，华林在陈述“华林定理”时，心中所想的就是这类结果。大卫·希尔伯特的证明引出了数论中若干重大的新定理。

华林还撰写了关于代数曲线的著作，将四次曲线分为12个主要类别和84551个细分。

与华林同时代的作者们对他的若干描述并不太恭维。有人写道，他：-

……是人类性格中虚荣与谦逊最强烈的结合之一。然而，前者是他的主要特征。

另一个人说他是：——

……英格兰所产生的最伟大的分析学家之一……[在他生命的最后阶段]陷入了近乎精神错乱的深深的宗教忧郁之中。

最后这一说法或许能部分解释这一奇怪的事实：尽管华林于1763年当选为皇家学会会士，并于1784年获得了被授予其⟦E1⟧奖章的极大殊荣，但他却在1795年以贫穷为由辞去了该学会的职务。不过，他还获得了其他荣誉，例如当选为哥廷根皇家学会和博洛尼亚皇家学会的会士。也许对华林最准确的评价是由Thomas Thomson作出的：——

华林是十八世纪最深邃的数学家之一；但他著作的不优雅和晦涩使他未能获得他本应享有的声誉。

Edward Waring's father, John Waring, was a farmer. Several generations of his family lived at Mytton in Shropshire. John Waring married Elizabeth and their son Edward was educated at Shrewsbury school.

Waring entered Magdalene College, Cambridge on 24 March 1753. He won a Milbridge scholarship and he was admitted as a sizar, meaning that he paid a reduced fee but essentially worked as a servant to make good the fee reduction. He immediately impressed his teachers with his mathematical ability and he graduated B.A. in 1757 as senior wrangler.

On 24 April 1758 Waring was elected a fellow of Magdalene College. Waring's most famous work was Miscellanea analytica de aequationibus algebraicis et curvarum proprietatibus Ⓣ which he worked on during the next few years. He submitted the first chapter of this work to the Royal Society but following this nothing happened for two years. When Waring was nominated for the Lucasian Chair of Mathematics at Cambridge in 1759, the work was distributed as Miscellanea Analytica Ⓣ to prove he was qualified for the post despite his youth.

William Powell of St John's College Cambridge had his own ideas about who should fill the Lucasian Chair of Mathematics and attempted to prevent Waring being appointed. He put out a pamphlet entitled Observations which criticised Waring and doubted his mathematical abilities. Waring responded to this criticism on 25 January 1760 with the pamphlet A reply to the observations. Powell, still following his own agenda, was not going to give up that easily and responded immediately with Defence of the observations. John Wilson now wrote A letter to support Waring and this was sufficient to see him confirmed as Lucasian professor on 28 January 1760 at the age of 23.

When Miscellanea Analytica Ⓣ was published as a complete work in 1762, Waring chose to call it a second edition. Rather strangely, despite it being a second edition, it was given a new title Meditationes Algebraicae Ⓣ. We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry. On 2 June 1763 Waring was elected a Fellow of the Royal Society.

In 1764 Lalande published Life of Condorcet. In this work it was claimed that there were no first-class analysts in England. Waring responded quickly to this comment by writing a letter to Nevil Maskelyne, the Astronomer Royal. In the letter Waring pointed out that d'Alembert, Euler and Lagrange had all praised his 1762 work. Waring wrote that in the book he had given:-

... somewhare between three and four hundred new propositions of one kind or another, considerably more than have been given by any other English writer.

However, Waring knew that his work had not been widely read for he added that he:-

... never could hear of any reader in England, out of Cambridge, who took pains to read and understand it ...

The reason that so few had read the book was partly because the subject matter was difficult, partly because Waring was a poor communicator, and partly because he did not have a good algebraic notation. It would be reasonable to compare Waring with Ruffini who, about 150 years later, suffered the same fate with his work in algebra for much the same reasons.

One would not expect the Lucasian professor of mathematics to take a medical degree but that is exactly what Waring did, graduating with his M.D. in 1767. For a short time he practised medicine in various London hospitals, then Addenbroke hospital in Cambridge and finally at a hospital in St Ives, Huntingtonshire. However, he gave up practising medicine by 1770 [8]:-

... he was very short-sighted and very shy in manner, so that he quickly abandoned his profession.

One might ask how Waring could practise medicine and hold the Lucasian Chair at the same time. Well, he never lectured as part of his duties. Some claim that it was because his ideas were so profound that they could not be communicated in lectures, but if truth be told it is more likely that the reason was because he was a poor communicator with handwriting which was almost impossible to read. [I will agree that such problems have not stopped others lecturing!]

In 1776 Waring married Mary Oswell. They lived for a while in Shrewsbury but the town was not to Mary's liking and the couple moved to Waring's estate at Plealey in Pontesbury.

Waring's Miscellanea Analytica ... Ⓣ of 1762 formed the basis, as we have noted, of further books. Proprietates algebraicarum curvarum Ⓣ, covering geometry, was published in 1772. Meditationes Algebraicae Ⓣ, covering the theory of equations and number theory, appeared in 1770 with an expanded version in 1782. A further work Meditationes analyticae Ⓣ appeared in 1776 with a new expanded edition in 1785.

In Meditationes Algebraicae Ⓣ Waring proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients. He derived a method for expressing symmetric polynomials and he investigated the cyclotomic equation $x^{n} - 1 = 0$. This work makes Waring one of the earliest contributers to Galois theory. In particular, discussing Problem 22 of Chapter 3, Weeks writes [3]:-

The most significant aspect of Waring's treatment of this example is the symmetric relation between the roots of the quartic equation and its resolvent cubic. This is, in essence, the first result in the theory of symmetric functions (beyond the basic building blocks which appeared in Chapter 1), a theory whose systematic development was not to appear until the 19th century (Lagrange, Gauss, and others) and was ultimately followed by the theory of permutation groups (Galois, Jordan, ...).

Chapter 4 of Meditationes Algebraicae Ⓣ contains results such as:-

k equations in k unknowns can be reduced to one equation with one unknown.

His result that the product of the degrees of the original equations is the degree of the single reduced equation is known as the Generalised Theorem of Bézout.

The rest of the book deals with number theory, a topic in which Waring made some interesting advances. He stated that any even integer can be written as the sum of two primes and every odd integer is either a prime or the sum of three primes. This result, now known as the Goldbach conjecture, is one of the most famous unsolved problems of mathematics. Although Goldbach proposed his question in a letter to Euler long before Waring published Meditationes Algebraicae Ⓣ it is still worth noting that Waring's version was the first to be published.

Waring also stated, without giving a proof, what is now known as 'Waring's theorem':-

Every integer is equal to the sum of not more than 9 cubes. Also every integer is the sum of not more than 19 fourth powers, and so on ....

In 1909 Hilbert proved that given any integer $n$ there is an integer $m$ (depending on $n$) such that every integer is a sum of $m$ $n$th powers. It is reasonable to assume that Waring had this type of result in mind when he stated 'Waring's theorem'. Hilbert's proof led to major new theorems in number theory.

Waring also wrote on algebraic curves, classifying quartic curves into 12 main divisions with 84551 subdivisions.

Several descriptions of Waring given by authors from his own period are not too flattering. One writes that he was:-

... one of the strongest compounds of vanity and modesty which the human character exhibits. The former, however, is his predominant feature.

Another says that he is:-

... one of the greatest analysts that England has produced ... [ near the end of his life being] sunk into a deep religious melancholy approaching to insanity.

This last statement may partly explain the strange fact that although Waring was elected a Fellow of the Royal Society in 1763 and had the great distinction of being awarded its Copley Medal in 1784, he resigned from the Society in 1795 claiming poverty. He was awarded other honours, however, such as election to the Royal Society of Göttingen and the Royal Society of Bologna. Perhaps the most accurate assessment of Waring was made by Thomas Thomson:-

Waring was one of the profoundest mathematicians of the eighteenth century; but the inelegance and obscurity of his writings prevented him from obtaining that reputation to which he was entitled.