威廉·金顿·克利福德William Kingdon Clifford

威廉·金顿·克利福德的父母是克利福德（1821-1878）和Fanny（Frances）克利福德（1821-1854）。克利福德 Sn.出生于赫特福德郡的Bishops Stortford，是一名书商，主要经营宗教材料，但也是埃克塞特的一位杰出公民，担任过市议员和治安法官。Fanny 克利福德是Francis Wilmott 克利福德和Mary-Anne Bodley的女儿，后者与Sir Thomas Bodley（1545-1613）有亲属关系。Sir Thomas曾是牛津大学Magdalen学院的自然哲学讲师，也是Bodleian图书馆的创始人。人们认为，本传记的主人公克利福德从母亲那里继承了许多天赋。然而，这里也有不利的一面，因为他似乎也从母亲那里继承了不佳的身体状况，母亲于1854年去世，年仅35岁。她的儿子克利福德当时只有九岁。他的父亲再婚，与第二任妻子又生了四个孩子。克利福德 Sn.也去世得很早，于1878年去世，年仅58岁。克利福德在埃克塞特长大，最初住在Park Place 9号，也就是他母亲出生的那所房子。1851年他肯定还住在那里，但后来他们搬到了埃克塞特High Street 23号。

克利福德在学校表现出极大的潜力，在许多不同科目中获奖。他在埃克塞特文法学校学习了几个月，然后于1856年转到埃克塞特的Mansion House School。这所学校由一位Templeton先生管理，在1858年和次年，克利福德参加了剑桥大学地方考试，科目广泛，在大多数科目中表现出色。15岁时，他获得了伦敦国王学院普通文学与科学系的数学和古典学奖学金，并于1860年入学。在那里，他在数学以及古典学、英国文学和（也许出乎意料地）体操方面都很出色。在国王学院度过的三年中，他每年都获得数学奖学金和古典学奖学金。他还在第一年获得了神学奖，并在随后几年获得了英语语言Inglis奖学金和英语论文奖。克利福德的第一篇数学论文是在国王学院期间写的，题为The Analogues of Pascal's Theorem，于1863年发表在The Quarterly Journal of Pure and Applied Mathematics 上。

1863年9月，在去剑桥前几天，克利福德写了以下这封信，这封信告诉我们很多关于他十八岁时的情况[26]：-

我一直在德文郡北部徒步旅行，否则我早就写信了。到明天，我希望寄给你一些东西，并会尽力遵循你的善意建议；但我自己还是一个非常初级的读者。几乎从我开始放风筝的时候起（我还没有停止），我就一直在思考在风力作用下找到风筝线形状的问题。前几天粗略试验时，内在方程似乎并不太难获得；如果我得到任何结果，明天就寄给你。我一直试图构建机械方程的第二种解释，类似于切线坐标的解释，但迄今为止失败了。作为一个坚信符号二元性的人，我应该把完全失败视为我们符号系统错误的证明。你会觉得我试图获得一种成打发明问题的方法的幻想尝试很有趣。

1863年10月，18岁时，克利福德进入剑桥大学三一学院，获得了基础奖学金学习数学和自然哲学（物理学）。他不仅赢得了数学奖项，还因关于沃尔特·雷利爵士的演讲而获奖。1867年，他在期末考试中获得第二名数学荣誉学位考试一等及格者（Wrangler）和史密斯奖后，被授予数学和自然哲学学士学位（与许多其他在剑桥获得第二名的著名数学家一样，如开尔文和詹姆斯·克拉克·麦克斯韦）。这些杰出成就必须与他在剑桥期间经历的基督教信仰危机相对照。Monty Chisholm写道[12]：-

克利福德早期的高教会宗教信念和对天主教神学的信仰是他家庭背景和教育的自然结果，但是，随着对达尔文在《物种起源》中提出的思想的讨论激烈进行，克利福德发现他早期的正式基督教信仰正在被侵蚀。此外，在剑桥的早年，他受到进化论早期支持者赫伯特·斯宾塞著作的影响，并开始与托马斯·赫胥黎和弗瑞兹·约翰廷德尔建立友谊。很快，他就获得了作为一个大声疾呼且雄辩的所有神职人员和教条宗教反对者的声誉。

当时，所有剑桥学生每年都必须签署《三十九条信纲》，以申明对英格兰教会的效忠。克利福德于1863年和1864年签了字，但到1865年需要他签字时，他严重怀疑自己能否真诚地签署；然而，在为此事焦虑之后，他还是签了。可是，当1866年需要他签字时，他拒绝了。这本来应当终结他的剑桥生涯，但令人惊讶的是，一切照常继续，他于1868年当选为三一学院的研究员。他的成功使他在1867年的捐助者纪念典礼上被邀请发表演讲。他演讲的开头是：

思想若无自身之外的结果便无能为力：征服世界的思想不是静观的，而是行动的。而正是这种思想，我今天请你们崇敬。

他在这篇演讲中提出的思想得到进一步发展，成为1868年他在皇家学会发表的讲座On Some of the Conditions of Mental Development，当时他年仅23岁。在讲座中，他试图解释科学发现是如何产生的：

没有哪位科学发现者、诗人、画家或音乐家不会告诉你，他发现自己的发现、诗篇或画作早已现成——它来自外部，而不是他有意识地从内部创造出来的。

他的本科朋友之一弗雷德里克·波洛克在克利福德的Lectures and Essays导言中，提供了克利福德本科时期的更多细节。见THIS LINK。

1870年，克利福德参加了英国日食远征队，航行到意大利以从日食中获取科学数据。他经历了在西西里附近船只失事的不幸经历，但幸运地活了下来。关于这次远征的详细描述，特别是克利福德的贡献，见THIS LINK。

1871年，克利福德被任命为伦敦大学学院数学与力学讲席。我们注意到，鉴于他已经转而反对基督教，伦敦大学学院对他来说是一个理想的地方，因为与剑桥不同，伦敦大学学院不要求任何宗教承诺。1874年，他当选为皇家学会会士。他还是伦敦数学会的热情成员，该学会在伦敦大学学院举行会议。1874年是他遇到索菲亚·露西Jane Lane（1846年8月2日-1929年4月21日）的一年，她是约翰桑德斯·麦克兰恩的女儿。他们于1875年4月7日结婚，住在伦敦科尔维尔路一个简朴的家中。哲学家爱德华·克洛德在他的书Memories中写道：-

在克利福德家里，你一定会遇到值得认识的人。没有Smart Set来填补他们的空虚时间，用无聊的闲话浪费你的时间；没有自命不凡的人用他们的做作激怒你；没有学究用他们的学术含糊让你厌烦，只有一群想要互相见面并充分自由交谈的男女。

克利福德和Lucy Clifford有两个女儿，Ellen Clifford和Alice Margaret Clifford。克利福德去世后，Lucy继续写了十部小说、七部短篇小说集以及许多戏剧和文章。这些小说有争议，但大多数卖得很好。她的第一部小说Mrs Keith's Crime 在1885年被评为年度图书。

现在让我们简要看看克利福德的数学贡献。尽管他生命短暂——他去世时年仅33岁——但他写了大约50篇论文。受波恩哈德·黎曼和罗巴切夫斯基工作的影响，克利福德研究了非欧几何。1870年，他写了On the space theory of matter，其中他论证能量和物质只是空间曲率的不同类型。在这部作品中，他提出了将在阿尔伯特·爱因斯坦广义相对论中起基础作用的思想。然而，他只指出这种方法可能，但并未将项目推进到计算细节。此外，在未发表的笔记中，他描述了连接看似无关现象的重要性（见[10]）：-

在不同科学探究分支之间建立联系的重要性；这样的步骤具有科学革命的性质。引力中的例子将非常不同的现象连接在一起。表象表明，当前科学状态正处于这样一场革命的前夜，这场革命将把气体、液体和固体的物理性质、化学结合定律、电磁现象以及光的理论连接在一起。……因此，所指向的是动理论与波动理论之间的联系。

这些随想中的细节如此之少，以至于可以将它们视为对20世纪几项重要科学进展的预测。鉴于克利福德辉煌而非传统的思维，这并非不可能。克利福德将四元数（由威廉·哈密顿在克利福德出生前两年引入）推广到他所谓的双四元数，在他的论文Preliminary Sketch of Biquaternions（1873）中，并用它们研究非欧空间和某些曲面上的运动。这些现在被称为“克利福德-菲利克斯·克莱因空间”。他表明常曲率空间可以有几种不同的拓扑结构。克利福德还证明了波恩哈德·黎曼曲面在拓扑上等价于一个带孔的盒子。他在论文Applications of Grassmann's extensive algebra（1878）中引入了克利福德代数。注意，这里我们有克利福德统一数学思想的例子。

作为教师，克利福德的声誉极为卓越。一位学生在研究詹姆斯·艾沃里关于椭球吸引力的定理时遇到问题，描述了克利福德对他提问的回应：-

他没有任何图表或符号辅助，就描述了解决方案所依赖的几何条件，这些条件似乎在空间中清晰可见。不再有需要推导的结论，而是真实而明显的事实，只需被看到。

克利福德不仅是一位极具原创性的教师和研究者，他还是一位科学哲学家。他在其文章On the Nature of Things-in-Themselves（1878年）中为构成良知的元素创造了“心灵材料”这一短语。我们引用这篇文章[15]：-

正如我们所看到的，即使是最简单的感觉也是复杂的，这种元素我将称之为心灵材料。无机物质的一个运动分子不具有心灵或意识；但它拥有心灵材料的一小部分。当分子如此结合，以至于形成水母下侧的薄膜时，伴随它们的心灵材料元素如此结合，以至于形成感觉的微弱开端。当分子如此结合，以至于形成脊椎动物的脑和神经系统时，相应的心灵材料元素如此结合，以至于形成某种意识；也就是说，同时发生的复杂变化如此联系在一起，以至于一个的重复意味着另一个的重复。当物质采取活人脑的复杂形式时，相应的心灵材料采取人类意识的形式，具有智力和意志。

克利福德于1876年11月7日星期日下午向“星期日讲座协会”发表演讲时，讲授了Right and wrong: the scientific ground of their distinction。他用以下描述宣传这次讲座[4]：-

我们觉得偷窃或说谎是错误的，照顾家人是正确的；并且我们对自己的行为负责。我们将这些感觉的总和称为良知，或道德感。在这次讲座中，提议考虑科学方法能对这些事实给出什么解释。这是一种通过推理获取知识的方法；首先，基于自然一致性的假设，从现象推断现象；其次，基于他人有与我相似感觉的假设，推断与这些现象同时并作为其基础的心理事实。这些假设中的每一个都建立在道德基础上；以这种方式指导我们的信念是我们的责任。

我们已经看到，克利福德在许多方面都是一个非凡的人。让我们再稍微看看他的性格。Macfarlane[18]告诉我们：-

……他在外表、习惯和见解上都很古怪。

Monty Chisholm写道[12]：-

克利福德是一个极有吸引力的伙伴。他喜欢有人做伴、喜欢聚会和跳舞。他写童话，讲笑话。

一位本科同学对他的描述很有意思：-

他的整洁和灵巧非同寻常，但最引人注目的是，与他的体重相比，他的力量极大，这在一些练习中有所体现。有一次，他能用任意一只手在单杠上做引体向上，众所周知，这是力量的最大壮举之一。

他与刘易斯·卡罗尔一样，喜欢招待孩子们。虽然他在成功上从未能与刘易斯·卡罗尔的刘易斯·卡罗尔书相匹敌，但克利福德写了The Little People，这是一本为逗孩子们开心而写的童话故事集。他非常喜欢参加儿童聚会，并想出一些新点子来逗孩子们开心。

1876年，克利福德身体崩溃。这肯定因过度工作而加重，如果不是完全由它引起的话。他会白天从事教学和行政职责，然后整夜进行研究。在阿尔及利亚和西班牙度过的六个月让他充分恢复，得以恢复职责18个月，但也许不可避免，他再次崩溃。在地中海国家度过的一段时间对他的健康几乎没有帮助，在1878年末回到英格兰几个月后，他前往马德拉。期望的恢复从未实现，几个月后他去世了。

他的文集扉页上印有艾萨克·牛顿在谈到罗杰·柯特斯时首次说出的一句话：-

如果他活着，我们或许能知道些什么。

事实上，他的大部分著作都是在他去世后出版的。两卷本著作Elements of dynamics 的第一卷于1878年出版，第二卷在他去世后问世。Lectures and essays和Seeing and thinking于1879年出版。Common sense of the exact sciences由卡尔·皮尔逊完成，于1885年出版。Lectures and essays和Common sense of the exact sciences的书评摘录见THIS LINK。

卡尔·皮尔逊为克利福德的Common sense of the exact sciences所写引言的摘录见THIS LINK。

让我们把关于克利福德的最后几句话留给他的妻子露西。1913年，当有人就Dictionary of National Biograph中克利福德的条目与她接洽时，Lucy Clifford回答说：-

如果你能再多说几句关于我丈夫的话，我会很高兴——他多么才华横溢，多么机智风趣，性格多么可爱：他是那么快乐、单纯、无忧无虑，还有一种难以形容的魅力。……在他去世时，人们说他是欧洲第三位数学家，阿瑟·凯莱和詹姆斯·约瑟夫·西尔维斯特位列前两位。他是他那个时代最杰出的三一学院研究员；Royal Society最年轻的研究员。如果他活着，没有什么是他做不到或不愿做的，因为生活的任何方面都对他有吸引力。我很喜欢你所说的，我的家是文学圣地之一。但你应该在他生前来到这里，各种阶层、各种思想的人都来过，无论他们与他多么意见相左，他们总是会一时被他迷住：一方面，他有一双奇妙的深色睫毛的蓝眼睛，似乎有一种奇妙的灵魂从中向外凝视。

为纪念克利福德，已组织了几场活动。其中一场于1995年举办，以庆祝克利福德诞辰150周年：-

在克利福德诞辰150周年之际，克利福德与Lucy Clifford研究小组将于1995年5月6日星期六在肯特大学卢瑟福学院举办为期一天的会议，以纪念Clifford夫妇。会议将穿插关于克利福德和Lucy的生平评述，以及知名学者在数学、科学、哲学和文学等主题上的专题报告，包括几何学与物理学的关系、科学与宗教的达尔文争议，以及数学与科学教育。

William Clifford's parents were William Clifford (1821-1878) and Fanny (Frances) Kingdon (1821-1854). William Clifford Sn., born in Bishops Stortford, Hertfordshire, was a book seller, mainly of religious material, but also a prominent citizen of Exeter being an Alderman and a Justice of the Peace. Fanny Kingdon was the daughter of Francis Wilmott Kingdon and Mary-Anne Bodley who was related to Sir Thomas Bodley (1545-1613). Sir Thomas had been a lecturer in natural philosophy at Magdalen College, Oxford and a founder of the Bodleian Library. It is thought that William Clifford, the subject of this biography, inherited much of his genius from his mother. However, there was a down-side here for he also appears to have inherited poor physical health from his mother who died in 1854 at the age of 35. Her son William was only nine years old at the time. His father remarried and had four more children with his second wife. William Sn. died quite young too, at the age of 58 in 1878. William was brought up in Exeter where first he lived at 9 Park Place, in the house in which his mother had been born. He was certainly living there in 1851 but later they moved to 23 High Street, Exeter.

William showed great promise at school where he won prizes in many different subjects. He studied at Exeter Grammar School for a few months before moving in 1856 to the Mansion House School in Exeter. This school was run by a Mr Templeton and in both 1858 and in the following year, Clifford sat the Cambridge University Local Examinations in a wide range of subjects, performing excellently in most of them. At age 15 he won a Mathematical and Classical Scholarship to the Department of General Literature and Science at King's College, London, which he entered in 1860. There he excelled in mathematics and also in classics, English literature and (perhaps unexpectedly) in gymnastics. In each of the three years he spent at King's College he won the Mathematical Scholarship and the Classical Scholarship. He also won the Divinity Prize in his first year and in subsequent years he won the Inglis Scholarship for English Language and the English Essay Prize. Clifford's first mathematics paper, written while he was at King's College, was The Analogues of Pascal's Theorem and it was published in The Quarterly Journal of Pure and Applied Mathematics in 1863.

A few days before he went up to Cambridge, in September 1863, Clifford wrote the following letter which tells us much about him at age eighteen [26]:-

I have been on a walking tour in the North of Devon, or I should have written long before. By to-morrow I hope to send you something, and will do what I can to follow out your kind suggestions; but I am a very junior reader myself. I have had in my mind almost from the time I began to fly kites (I have not yet left off) the problem of finding the form of a kite-string under the action of the wind. On a rough trial the other day, the intrinsic equation seemed not very difficult to obtain; if I get at any result, I will send it you to-morrow. I have been trying to construct a second interpretation of mechanical equations, similar to that of tangential coordinates, but have failed hitherto. Being a firm believer in the duality of symbols, I should look upon complete failure as a proof that our symbolical system is wrong. You will be amused by my visionary attempt at obtaining a method of inventing problems by the dozen.

When he was 18 years old, in October 1863, William entered Trinity College, Cambridge, having won a Foundation Scholarship to study mathematics and natural philosophy (physics). He won not only prizes for mathematics but also a prize for a speech he delivered on Sir Walter Raleigh. In 1867 he was awarded a BA in Mathematics and Natural Philosophy after being Second Wrangler and Smith's Prizemen in his final examinations (in common with many other famous mathematicians who were second at Cambridge like Thomson and Maxwell). These outstanding achievements must be set against a crisis in his Christian faith which he experienced while at Cambridge. Monty Chisholm writes [12]:-

Clifford's early High Church religious convictions and belief in Catholic theology were a natural result of his family background and teaching but, as discussions raged over the ideas propounded by Darwin in 'The Origin of Species', Clifford found his early formal Christian faith becoming eroded. Moreover, in his early years at Cambridge, he was influenced by the writings of the early proponent of evolutionary theory, Herbert Spencer, and he began his friendships with Thomas Huxley and John Tyndall. Very soon, he was acquiring a reputation as a vociferous and eloquent opponent of all priesthoods and dogmatic religions.

At that time all Cambridge students had to sign the Thirty-nine Articles of Protestant Faith each year affirming their allegiance to the Church of England. Clifford signed in 1863 and 1864, but by the time he was required to sign in 1865 he seriously doubted whether he could do so in good faith but, after worrying about the act, nevertheless did sign. However, when he was required to sign in 1866, he refused. This should have brought his Cambridge career to an end but, surprisingly, things continued as normal and he was elected to a Fellowship at Trinity in 1868. His successes had led to him being asked to give the speech at the Commemoration of Benefactors Ceremony in 1867. The opening words of his address were:-

Thought is powerless except it makes something outside of itself: the thought which conquers the world is not contemplative but active. And it is this that I am asking you to worship today.

The ideas that he presented in this address he developed further and this became the lecture On Some of the Conditions of Mental Development given to the Royal Institution in 1868 when he was only 23 years old. In the lecture he tried to explain how scientific discovery comes about:-

There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture - that it came to him from outside, and that he did not consciously create it from within.

Frederick Pollock, one of his undergraduate friends, gives further details of Clifford's time as an undergraduate in his Introduction to Clifford's Lectures and Essays. See THIS LINK.

In 1870 Clifford was part of the English Eclipse Expedition which sailed to Italy to obtain scientific data from an eclipse. He had the unfortunate experience of being shipwrecked near Sicily, but he was fortunate to survive. For a detailed description of this Expedition, and particularly Clifford's contributions, see THIS LINK.

In 1871 Clifford was appointed to the chair of Mathematics and Mechanics at University College, London. We note that, given the fact that he had turned against Christianity, University College was the ideal place for him since, unlike Cambridge, University College did not require any religious commitment. In 1874 he was elected a Fellow of the Royal Society. He was also an enthusiastic member of the London Mathematical Society which held its meetings at University College. The year 1874 was the one in which he met Sophia Lucy Jane Lane (2 August 1846-21 April 1929), the daughter of John Lane. They were married on 7 April 1875 and lived in a modest home on Colville Road, London. The philosopher Edward Clodd wrote in his book Memories:-

At the Clifford's you were sure to meet someone worth the knowing. There was no Smart Set to fill their empty time and waste yours in inane gossip; no prigs to irritate you with their affectation; no pedants to bore you with their academic vagueness, but just a company of men and women who wanted to meet one another and have a full and free talk.

William and Lucy Clifford had two daughters, Ellen Clifford and Alice Margaret Clifford. After Clifford's death, Lucy went on to write ten novels, seven collections of short stories and many plays and articles. The novels were controversial but most sold well. Her first novel Mrs Keith's Crime was voted book of the year in 1885.

Let us now look briefly at Clifford's mathematical contributions. Despite his short life - he died aged 33 - he wrote around 50 papers. Influenced by the work of Riemann and Lobachevsky, Clifford studied non-euclidean geometry. In 1870 he wrote On the space theory of matter in which he argued that energy and matter are simply different types of curvature of space. In this work he presents ideas which were to form a fundamental role in Einstein's general theory of relativity. However, he only pointed out that such an approach was possible but he did not carry the project as far as working out details. Also in unpublished notes he described the importance of connecting apparently unrelated phenomena (see [10]):-

Importance of establishing connections between different branches of scientific enquiry; such steps are of the nature of revolutions in science. Example in gravitation which connected together very various phenomena. Appearances indicate that present state of science is on the eve of such a revolution, which will connect together physical properties of gases, liquids and solids, laws of chemical combination, electromagnetic phenomena, and the theory of light. ... What is pointed to is therefore a connection between kinetic theory and undulatory theory.

There is so little detail in these random thoughts that it is possible to see them as predicting several important scientific advances of the 20th century. Given Clifford's brilliant and unconventional thinking, this is not impossible. Clifford generalised the quaternions (introduced by Hamilton two years before Clifford's birth) to what he called the biquaternions in his paper Preliminary Sketch of Biquaternions (1873) and he used them to study motion in non-euclidean spaces and on certain surfaces. These are now known as 'Clifford-Klein spaces'. He showed that spaces of constant curvature could have several different topological structures. Clifford also proved that a Riemann surface is topologically equivalent to a box with holes in it. He introduced Clifford algebras in his paper Applications of Grassmann's extensive algebra (1878). Note that here we have examples of Clifford unifying ideas in mathematics.

As a teacher Clifford's reputation was outstanding. A student, having problems with Ivory's theorem on the attraction of an ellipsoid, describes Clifford's response to his questions:-

Without any diagram or symbolic aid he described the geometrical conditions on which the solution depended, and they seemed to stand out visibly in space. There were no longer consequences to be deduced, but real and evident facts which only required to be seen.

Not only was Clifford a highly original teacher and researcher, he was also a philosopher of science. He coined the phrase 'mind-stuff' for the elements from which conscience is composed in his article On the Nature of Things-in-Themselves (1878). We quote from this article [15]:-

That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mind-stuff. A moving molecule of inorganic matter does not possess mind or consciousness; but it possesses a small piece of mind-stuff. When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff which go along with them are so combined as to form the faint beginnings of Sentience. When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mind-stuff are so combined as to form some kind of consciousness; that is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other. When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition.

Clifford delivered the lecture Right and wrong: the scientific ground of their distinction when he addressed the 'Sunday Lecture Society' on the afternoon of Sunday 7 November 1876. He advertised the lecture with a description that began as follows [4]:-

We feel that it is wrong to steal or tell lies, and right to take care of our families; and that we are responsible for our actions. The aggregate of such feelings we call Conscience, or the Moral Sense. In this lecture it is proposed to consider what account can be given of these facts by the scientific method. This is a method of getting knowledge by inference; first of phenomena from phenomena on the assumption of uniformity of nature, and secondly of mental facts simultaneous with and underlying these phenomena, on the assumption that other men have feelings like mine. Each of these assumptions rests on a moral basis; it is our duty to guide our beliefs in this way.

We have already seen that Clifford was a remarkable man in many ways. Let us look a little more at his character. Macfarlane [18] tells us that:-

... he was eccentric in appearance, habits and opinions.

Monty Chisholm writes [12]:-

Clifford was a most attractive companion. He loved company and parties and dancing. He wrote fairy tales and told jokes.

A description of him by a fellow undergraduate is interesting:-

His neatness and dexterity were unusually great, but the most remarkable thing was his great strength as compared with his weight, as shown in some exercises. At one time he would pull up on the bar with either hand, which is well known to be one of the greatest feats of strength.

He shared with Charles Dodgson the pleasure of entertaining children. Although he never rivalled Dodgson's Lewis Carroll books in success, Clifford wrote The Little People, a collection of fairy stories written to amuse children. He greatly enjoyed going to children's parties and thinking up some new idea to amuse the children.

In 1876 Clifford suffered a physical collapse. This was certainly made worse by overwork if not completely caused by it. He would spend the day with teaching and administrative duties, then spend all night at his research. Six months spent in Algeria and Spain allowed him to recover sufficiently to resume his duties for 18 months but, perhaps inevitably, he again collapsed. A period spent in Mediterranean countries did little for his health and after a couple of months back in England in late 1878 he left for Madeira. The hoped-for recovery never materialised and he died a few months later.

The title page of his collected works contains the quote first made by Newton speaking of Cotes:-

If he had lived we might have known something.

Most of his work was in fact published after his death. Volume 1 of the two volume work Elements of dynamics was published in 1878, the second volume appearing after his death. Lectures and essays and Seeing and thinking were published in 1879. Common sense of the exact sciences was completed by Karl Pearson and published in 1885. Extracts from reviews of Lectures and essays and Common sense of the exact sciences are given at THIS LINK.

An extract from Karl Pearson's Introduction to Clifford's Common sense of the exact sciences is given at THIS LINK.

Let us leave the final words about Clifford to his wife Lucy. When approached in 1913 about an entry for Clifford in the Dictionary of National Biography, Lucy Clifford replied:-

If you can say a few more words about my husband I would love it - how brilliant he was, how witty and what an adorable nature he had: he was so gay and simple and light-hearted, and had an indescribable charm. ... At the time of his death he was said to be the third mathematician in Europe, Cayley and Sylvester being the two first. He was the most brilliant Fellow of Trinity of his day; the youngest fellow of the Royal Society. There was nothing that he couldn't have done or wouldn't have done if he had lived, for there was no side of life that did not appeal to him. I love what you say about my house being one of the Meccas of Literature. But you should have come here in his time, people of every rank and of every shade of thought came, and no matter how little they agreed with him, they were always hypnotised for the hour: for one thing he had such wonderful dark-lashed blue eyes and a wonderful soul seemed to be looking out of them.

Several events have been organised to honour Clifford's memory. One was organised in 1995 to celebrate the 150th anniversary of Clifford's birth:-

On the occasion of the 150th anniversary of William Clifford's birth, the William and Lucy Clifford Research Group is hosting a one day meeting on Saturday 6 May 1995, at Rutherford College, University of Kent, in honour of the Cliffords. A biographical commentary on William and Lucy will be interspersed with specialist contributions from eminent academics on topical mathematical, scientific, philosophical and literary subjects including the relationship between geometry and physics, the Darwinian controversy over science and religion, and mathematical and scientific education.