阿瑟·凯莱Arthur Cayley

阿瑟·凯莱的父亲Henry Cayley（1768-1850）虽然出身于一个在英格兰约克郡生活了许多代的家族，却在俄罗斯圣彼得堡做商人。Henry Cayley娶了威廉·道蒂的女儿玛丽亚·安东尼娅·道蒂（1794-1875）。亨利和Maria Cayley有五个孩子：索菲亚·凯莱（1816-1889）、威廉·Henry Cayley（1818-1819）、本传记的主人公凯莱、查尔斯·巴戈特·凯莱（1823-1883）和亨丽埃塔-卡罗琳·凯莱（1828-1886）。从这些日期可以看出，他们的长子威廉·亨利在婴儿期就夭折了。与圣彼得堡的联系不仅仅是Henry Cayley的工作把他带到那里，因为凯莱的祖父、John Cayley（1730-1795）曾在圣彼得堡担任总领事。这个家族虽然住在圣彼得堡，但夏天会回英格兰，1821年的一次夏季探访中凯莱出生了。他的弟弟查尔斯·巴戈特出生在俄罗斯，后来以翻译但丁和荷马而闻名。凯莱在圣彼得堡度过了生命的最初七年，在那里接触了多种语言，尤其是俄语、英语和法语——那里的国际商务语言是法语。1828年，全家回到英格兰永久定居，住进了伦敦摄政公园附近约克 terrace 29号的一栋好房子里，此时六十岁的亨利成为伦敦 assurance 公司的董事。正是在全家回到伦敦后，凯莱的妹妹亨丽埃塔-卡罗琳在那里出生。在描述凯莱家族时，我们应该提到，皇家学会会士George Cayley爵士（1773-1857），航空导航和航空工程的先驱，是凯莱的第四代堂亲。然而，没有证据表明他们之间有联系。

凯莱在布莱克希思的一所私立学校就表现出高超的数值计算能力，1835年他转入国王学院学校时年仅14岁，而非通常的入学年龄16岁，他对高等数学的天赋变得明显。然而，他不仅擅长数学，还在许多科目中获奖。特别是，尽管没有专攻科学，他在最后两年都获得了化学奖。他的数学老师建议他的父母鼓励凯莱在这一领域深造，而不是遵从父亲的意愿进入家族企业做商人。

1838年，凯莱开始在剑桥大学三一学院学习，第一年由乔治·皮科克担任导师。他由威廉·霍普金斯辅导，后者鼓励他阅读大陆数学家的论文。他最喜欢的数学主题是线性变换和解析几何，还在本科时他就在新成立的Cambridge Mathematical Journal上发表了3篇论文，该刊由Duncan Gregory编辑。凯莱于1842年作为高级数学荣誉学位考试一等及格者（Wrangler）毕业，并获得了第一个史密斯奖。考试结束后，凯莱和他的朋友埃德蒙·维纳布尔斯带领一个本科生阅读小组前往苏格兰的阿伯费尔迪。其中一位本科生是弗朗西斯·高尔顿，他描述了他的导师凯莱（引自[11]）：-

从未有人的外在体格如此与其才能不符，正如凯莱那样。他的举止中有一种怪异而神秘的东西，使陌生人倾向于认为他既不完全神志健全，也没有多少才智，而这与事实恰恰相反……他看起来如此虚弱，以至于无法从事普通的体力劳动。

在旅行期间，Venables写信给剑桥的乔治·加布里埃尔·斯托克斯（引自[11]）：-

我们被苏格兰一些最高的山峰所环绕，其中最美的无疑是Schiehallion，它在引力史上很有名；我一直试图劝说凯莱在其山顶重复内维尔·马斯基林博士的实验，并为自己赢得不朽的名声，但徒劳无功；唉，他对名声毫无欲望，并且对各种实验和计算有着根深蒂固的厌恶：他现在坐在我身边，继续进行那些在《数学杂志》上开始的可怕研究，在其中用尽了希腊和英文字母表的所有字母后，他只好把δελτα颠倒过来，并求助于古英语。

凯莱在剑桥任教四年，获得了 fellowship，在此期间，他在Cambridge Mathematical Journal上发表了28篇论文。他研究了各种各样的数学主题，包括代数曲线和曲面、椭圆函数、行列式以及积分理论。例如，他在1843年发表了On a theory of determinants，其中他将二维行列式（行和列）的概念扩展到多维数组。1844年，他开始与乔治·布尔通信，这对两位数学家都很有价值。例如，他在1845年写信给乔治·布尔（当时乔治·布尔正在英格兰林肯的一所学校任教）：-

我希望我能设法去林肯一趟，我会非常享受与你讨论一些事情，更不用说你的大教堂的诱惑了。我想我必须在接下来的六个月里设法安排一下，尽管没有铁路，而如今人们开始认为自己有权享有铁路。

1844年，凯莱与他的朋友Edmund Venables一起前往瑞士阿尔卑斯山，还游览了意大利。他采取了比许多英国数学家同行更为国际化的方式，在法国Journal de Mathématiques Pures et Appliquées和奥古斯都·利奥波德·克雷勒的期刊Journal für die reine und angewandte Mathematik上发表论文。他在1845年和1846年发表的两篇论文被认为为不变量理论奠定了基础。

剑桥的会士职位任期有限，因为凯莱不打算接受圣职，所以他必须找一份职业。他选择了法律，并于1846年4月开始接受培训。还在接受律师培训期间，凯莱前往都柏林听威廉·哈密顿关于四元数的讲座。在这些讲座中，他坐在乔治·萨蒙旁边，两人在此后多年里交换数学思想。凯莱是威廉·哈密顿的好朋友，尽管两人在四元数在几何研究中的重要性上意见不一。凯莱的另一位朋友是詹姆斯·约瑟夫·西尔维斯特，他也从事法律职业。两人都在伦敦林肯律师学院工作，他们在工作日讨论深奥的数学问题。林肯律师学院中其他活跃的数学家包括休‧布来克本。他写信给William Thomson，谈到他与凯莱的工作：-

我们一直忙于（在我们本应起草议会法案和诸如此类尘世事务的时候）构造由直圆柱与半径为其两倍的球面交线的切线所生成的展曲面……

事实上，凯莱和开尔文也是好朋友。开始法律培训大约一年后，凯莱开始与托马斯·柯克曼通信。他是托马斯·柯克曼在Cambridge and Dublin Mathematical Journal 上那篇著名论文的审稿人，该论文展示了今天所谓的雅各布·施泰纳系统的存在性。凯莱在报告中写道：-

托马斯·柯克曼的论文显然很有趣，其主要结果非常优雅。

他于1849年5月3日取得律师资格。他当了14年律师，但凯莱尽管非常精通产权转让（他的法律专长），却始终将其视为赚钱的手段，以便能够追求数学。G·B·豪斯泰德写道[21]：-

法律对他而言始终是苦差事。法律形式中过多冗长的辞藻总是令他反感。他曾评论说：“法律的目标是用最多的词语说一件事，数学则是用最少的词语说它。”

在这14年的律师生涯中，凯莱发表了约250篇数学论文——有多少全职数学家能比得上这位“业余爱好者”的产出？然而，Tony Crilly写道[11]：-

在剑桥，甚至作为一名见习律师，他都可以退回到自己的书房。作为一名新获得资格的律师，随着科学声誉的日益增长，他不得不采取一种更为公开的形象，这一角色与他退隐的性格并不相称。凯莱与剑桥保持联系，并从担任三一学院年度考试的高级考官，晋升为1851年数学荣誉学位考试的高级主考人和1852年的高级考官。

因此，他开始寻找学术职位，为了提高获得教授职位的几率，他开始提高本已极高的数学论文发表率。例如，在1853-1856年间，他平均每年发表十篇论文，但为了提升自己的知名度，他在1857-1860年间平均每年发表三十篇论文。他在1856年申请阿伯丁马里沙尔学院自然哲学讲席时未能成功。他还对1858年剑桥的朗迪安几何与天文学讲席以及次年格拉斯哥大学的天文学讲席感兴趣。然而，他未能获得这两个职位以及其他职位。也许一个原因是，尽管有出色的研究记录，但他几乎没有教学经验。

1863年，凯莱被任命为剑桥大学纯数学萨德莱里安教授。尽管还有其他高素质的申请人，如艾萨克·托德夯特、Norman Ferrers和爱德华·约翰·劳思，他仍然成功了。这一任命使凯莱的收入大幅减少，他现在只能靠一份薪水生活，而这只是他作为熟练律师所赚收入的一小部分。然而，凯莱非常高兴有机会全身心投入数学。作为纯数学萨德莱里安教授，他的职责是：-

……解释和教授纯数学的原理，并致力于推动这门科学的发展。

凯莱将超额完成这些条件。然而，在我们简要回顾他卓越的研究成就之前，我们注意到他在1863年结婚，也就是他被任命为萨德莱里安教授的那一年。9月8日，他与来自格林威治的Susan Moline（1831-1923）结婚，她是Robert Moline的女儿。凯莱和Susan 凯莱有两个孩子：Henry Cayley（1870-1949），他在剑桥学习数学，但认为自己无法达到父亲的成就，于是成为了一名建筑师；以及Mary Cayley（1872-1950）。1864年，凯莱购买了位于剑桥小圣玛丽桑德斯·麦克兰恩的Garden House。1863年11月8日，凯莱发表了他的就职演讲Analytical geometry - Introductory lecture。然而，他随后几年的讲座往往基于他最新的研究，学生们发现这些讲座与荣誉学位考试无关，而他们都旨在获得尽可能高的分数[10]：-

凯莱关注的是数学“本身”，而大多数学生则希望在限定的教学大纲内具备能力。任何偏离大纲的做法都被视为浪费时间和精力。

他发表了900多篇论文和札记，几乎涵盖了现代数学的每一个方面。他最重要的工作是发展了矩阵代数、非欧几何以及$n$维几何。关于他在矩阵方面的工作，Richard Feldmann写道[17]：-

尽管“矩阵”一词是由詹姆斯·约瑟夫·西尔维斯特于1850年引入数学文献的，但创立矩阵理论的功劳必须归于凯莱，因为他发表了关于该主题的最早的阐述性文章。……凯莱关于矩阵理论的导论性论文用法文写成，发表在德文期刊上[1855年]。……[他]相当粗略地引入了逆矩阵和矩阵乘法的概念，或者如凯莱所称的“复合”。上述基本性质在第二篇阐述性文章[1858年]中得到扩展，该文还列出了矩阵的许多其他性质。

早在1849年，凯莱就写了一篇论文，将他关于置换的思想与奥古斯丁·路易·柯西的思想联系起来。1854年，凯莱写了两篇论文，因其对抽象群的洞察力而引人注目。当时已知的群只有置换群，而即便这还是一个全新的领域，凯莱却定义了抽象群，并给出了一个表来展示群的乘法。他给出了一些特殊置换群的“凯莱表”，但更重要的是，对于引入抽象群概念而言，他意识到矩阵和四元数是群。以下是他对抽象群定义的一部分：-

……符号1自然表示一种运算，它（无论是普遍地还是就特定的运算对象而言）使运算对象保持不变。……符号xy表示复合运算，执行该运算等价于先执行运算y，然后执行运算x；xy当然一般不同于yx。但符号x、y、……一般满足x(yz) = (xy)z等，因此xyz、xyzt等具有确定的意义，与复合这些符号的具体方式无关……

凯莱发展了代数不变理论，他发展的$n$维几何已被应用于物理学中研究时空连续统。他在矩阵方面的工作为量子力学奠定了基础，量子力学由维尔纳·海森堡于1925年发展起来。凯莱还提出欧几里得几何和非欧几何是几何的特殊类型。他将射影几何与依赖于角度大小和线段长度的度量几何统一起来。

你可以在THIS LINK上看到G·B·豪斯泰德对凯莱论文的评论。

他只出版了一本书，即An Elementary Treatise on Elliptic Functions（1876年）。然而，他为彼得·格思里·泰特的An Elementary Treatise on Quaternions （1890年）撰写了第6章，并出版了六便士小册子The Principles of Book-Keeping by Double Entry（1894年）。

1881年，他受邀在美国约翰威廉·霍普金斯大学讲授一门课程，他的朋友詹姆斯·约瑟夫·西尔维斯特在那里担任数学教授。1882年1月至5月，他在约翰霍普金斯大学讲授阿贝尔函数和Theta函数。回到英国后，Royal Society授予他科普利奖章。1883年，凯莱成为英国协会 for the Advancement of Science主席。在主席致辞凯莱中，他对自己数学观点作了初等阐述。他对几何学的观点是：-

众所周知，欧几里得的第十二公理，即使采用约翰·普莱费尔的形式，也被认为需要证明；而罗巴切夫斯基构造了一个完全一致的理论，其中假定该公理不成立，或者说一个非欧平面几何体系。我自己的观点是，欧几里得的第十二公理在约翰·普莱费尔的形式中不需要证明，而是我们经验的一部分——即我们通过经验所认识的空间，但它是作为一切外部经验基础的表象。波恩哈德·黎曼的观点……是，我们在‘理智中’有一个更一般的空间概念（事实上是非欧空间的概念），我们通过经验得知空间（我们经验的物理空间）即使不是精确地，至少也是以最高程度的近似，是欧几里得空间。但假设我们经验的物理空间只是近似地是欧几里得空间，会得出什么结论？不是几何命题只是近似为真，而是它们对于那个长期以来被视为我们经验的物理空间的欧几里得空间来说，仍然绝对为真。

关于凯莱的两段描述，都是他老年时的样子，很有意思。Macfarlane[15]说：

……我参加了伦敦数学会的一次会议。房间很小，大约十二位数学家围桌而坐，其中有凯莱教授……会议结束时，凯莱热情地与我握手，并对他读过的我的论文给予了最亲切的评价。他当时大约60岁，明显驼背，衣服显得宽大。他最引人注目的是他灰色眼睛的活跃目光和他特有的孩子气的微笑。

他的朋友之一托马斯·阿彻·赫斯特写道：

……一个瘦弱、看起来虚弱的人，头大，脸上有天花疤痕：他说话困难，略有口吃。他从不坐直在椅子上，而是臀部只挨着椅子边缘，一只肘撑在椅座上，另一只手臂搭在椅背上。

以下是G·B·豪斯泰德的悼词[23]：-

凯莱除了他惊人的独创性之外，无疑是最博学多才的数学家。在科学方面，可以说他无所不知，而且他是最后一个能够无所不知的人。他性格非常温和、亲切。詹姆斯·约瑟夫·西尔维斯特告诉我，他只见过他发过一次怒，那是在（两人都在从事法律工作！）一个信使打断他们的会面，送来一大堆法律文件——凯莱的新业务。凯莱一时厌恶，把文件摔在地上。

我们最后给出[8]的悼词：-

凯莱是十九世纪英国杰出的纯数学家。作为代数学家、分析学家和几何学家，他能够将这些广阔的研究领域联系起来。超过五十个数学概念和定理以他的名字命名。到他生命的尽头，他受到全世界数学家的崇敬。

除了我们上面提到的科普利奖章外，我们注意到凯莱于1852年当选为皇家学会会士，并于1859年获得该学会的皇家奖章。London Mathematical Society于1884年授予他奥古斯塔斯·德摩根奖章。他于1868年至1870年担任该学会主席。他于1865年当选为爱丁堡皇家学会会士。他于1857年当选为皇家天文学会会士，并于1859年至1881年担任该学会出版物的编辑。1872年，他被授予剑桥大学三一学院荣誉院士。他获得了剑桥、牛津、爱丁堡、都柏林、哥廷根、海德堡、莱顿和博洛尼亚大学的荣誉学位。月球上的一座环形山以凯莱命名。他是法国研究院的外籍荣誉院士，并当选为柏林、哥廷根、圣彼得堡、米兰、罗马、莱顿、乌普萨拉和匈牙利科学院的院士。

经过长期病痛后，他在剑桥的家中——花园屋去世。他于1895年2月2日葬于剑桥米尔路公墓。许多外国代表以及许多英国主要学者参加了他的葬礼。

Arthur Cayley's father Henry Cayley (1768-1850), although from a family who had lived for many generations in Yorkshire, England, worked as a merchant in St Petersburg, Russia. Henry Cayley was married to Maria Antonia Doughty (1794-1875), a daughter of William Doughty. Henry and Maria Cayley had five children: Sophia Cayley (1816-1889), William Henry Cayley (1818-1819), Arthur Cayley the subject of this biography, Charles Bagot Cayley (1823-1883) and Henrietta-Caroline Cayley (1828-1886). As is evident from these dates, their eldest child William Henry died as an infant. The connection with St Petersburg was more than just where Henry Cayley's job had taken him for his father, Arthur Cayley's paternal grandfather, John Cayley (1730-1795), had served as Consul General in St Petersburg. The family, although living in St Petersburg, returned to England for the summers and it was on such a summer visit in 1821 that Arthur Cayley was born. His younger brother Charles Bagot was born in Russia and went on to distinguish himself as translator of Dante and Homer. Arthur spent the first seven years of his life in St Petersburg where he came in contact with several languages, particularly Russian, English and French - the international business language there was French. The family returned to live permanently in England in 1828 and took up residence in a fine house at 29 York Terrace near Regent's Park in London, where Henry, now aged sixty, became a director of the London Assurance Corporation. It was after the family returned to London that Arthur Cayley's sister Henrietta-Caroline was born there. While we are describing the Cayley family we should mention that Sir George Cayley, F.R.S. (1773-1857), a pioneer of aerial navigation and aeronautical engineering, was Arthur Cayley's fourth cousin. There is, however, no evidence of contact between them.

Arthur showed great skill in numerical calculations at a private school in Blackheath and, after he moved to King's College School in 1835, at age 14 rather than the usual age of entry of 16, his aptitude for advanced mathematics became apparent. However, it was not only mathematics at which he excelled, for he won prizes in many subjects. In particular, he won the Chemistry Prize in each of his final two years despite not specialising in science. His mathematics teacher advised his parents that Arthur be encouraged to pursue his studies in this area rather than follow his father's wishes to enter the family business as a merchant.

In 1838 Arthur began his studies at Trinity College, Cambridge, having George Peacock as tutor in his first year. He was coached by William Hopkins who encouraged him to read papers by continental mathematicians. His favourite mathematical topics were linear transformations and analytical geometry and while still an undergraduate he had three papers published in the newly founded Cambridge Mathematical Journal edited by Duncan Gregory. Cayley graduated as Senior Wrangler in 1842 and won the first Smith's prize. After the examinations, Cayley and his friend Edmund Venables led a reading party of undergraduates to Aberfeldy in Scotland. One of these undergraduates was Francis Galton who described his tutor Cayley (quoted in [11]):-

Never was a man whose outer physique so belied his powers as that of Cayley. There was something eerie and uncanny in his ways, that inclined strangers to pronounce him neither to be wholly sane nor gifted with much intelligence, which was the very reverse of the truth ... he appeared so frail as to be incapable of ordinary physical work.

While on the trip, Venables wrote to George Stokes in Cambridge (quoted in [11]):-

We are surrounded by some of the highest mountains in Scotland, among which the most beautiful is certainly Schiehallion famous in the history of gravitation; I have been trying in vain to induce Cayley to repeat Dr Maskelyne's experiments on its summit and to acquire to himself a never dying fame; but alas he has no desire for notoriety, and has a rooted aversion to experiments and calculations of all kinds: he is now sitting by my side carrying on those dreadful investigations commenced in the Mathematical Journal, in which having exhausted all the letters of the Greek and English alphabets, he is fain to turn his δελτα s tops turvy, and have recourse to the old English.

For four years Cayley taught at Cambridge having won a Fellowship and, during this period, he published 28 papers in the Cambridge Mathematical Journal. He worked on a large variety of mathematical topics including algebraic curves and surfaces, elliptic functions, determinants and the theory of integration. For example he published On a theory of determinants in 1843 in which he extended the idea of a 2-dimensional determinant (rows and columns) to multidimensional arrays. In 1844 he began a correspondence with George Boole which proved valuable to both mathematicians. For example he wrote to Boole in 1845 (Boole was teaching at a school in Lincoln, England, at the time):-

I wish I could manage a visit to Lincoln, I should so much enjoy talking over some things with you, not to mention the temptation of your Cathedral. I think I must contrive it some time in the next six months, in spite of there being no railway, which one begins to consider oneself entitled to in these days.

In 1844 Cayley, along with his friend Edmund Venables, made a trip to the Swiss Alps, also visiting Italy. He took a much more international approach than many of his fellow English mathematicians, publishing papers in the French Journal de Mathématiques Pures et Appliquées and in Crelle's journal Journal für die reine und angewandte Mathematik. Two papers he published in 1845 and 1846 are regarded as laying the foundations for invariant theory.

The Cambridge fellowship had a limited tenure, since Cayley was not prepared to take Holy Orders, so he had to find a profession. He chose law and began training in April 1846. While still training to be a lawyer Cayley went to Dublin to hear William Rowan Hamilton lecture on quaternions. He sat next to George Salmon during these lectures and the two were to exchange mathematical ideas over many years. Cayley was a good friend of Hamilton's although the two disagreed as to the importance of the quaternions in the study of geometry. Another of Cayley's friends was James Joseph Sylvester who was also in the legal profession. The two both worked at the courts of Lincoln's Inn in London and they discussed deep mathematical questions during their working day. Others at Lincoln's Inn who were active mathematicians included Hugh Blackburn. He wrote to William Thomson about his work with Cayley:-

We have been busy (when we ought to have been drawing acts of parliament and such sublunary matters) in constructing the developable surface generated by the tangent to the curve of intersection of the straight cylinder and sphere of double its radius ...

In fact Cayley and Thomson were also good friends. About a year after he began legal training Cayley began a correspondence with Thomas Kirkman. He was a referee of Kirkman's famous paper in the Cambridge and Dublin Mathematical Journal in which he shows the existence of what today are called Steiner systems. Cayley wrote in his report:-

Kirkman's paper is decidedly interesting and his main result a very elegant one.

He was admitted to the bar on 3 May 1849. He spent 14 years as a lawyer but Cayley, although very skilled in conveyancing (his legal speciality), always considered it as a means to make money so that he could pursue mathematics. George Halsted writes [21]:-

The law was always drudgery to him. The superabundant verbiage of legal forms was always distasteful to him. He once remarked that "the object of law was to say a thing in the greatest number of words, of mathematics to say it in the fewest."

During these 14 years as a lawyer Cayley published about 250 mathematical papers - how many full time mathematicians could compare with the productivity of this 'amateur'? However, Tony Crilly writes [11]:-

At Cambridge, and even as a pupil barrister, he could retire to his study. As a newly qualified barrister, with a growing scientific reputation, it became obligatory to adopt a more public persona, a role which did not sit easily with his retiring disposition. Cayley maintained contact with Cambridge and from serving as Senior Examiner at the annual Trinity College examinations, he progressed to the responsibility of being Senior Moderator for the Mathematical Tripos in 1851 and Senior Examiner in 1852.

As a result he began to look for academic appointments and, to improve his chances of a professorship he began to increase his already remarkably high rate of publication of mathematics papers. For example, during the years 1853-1856 he averaged ten published papers per year, but as he sought to raise his profile, he published on average thirty papers per year during 1857-1860. He had not been successful in his application in 1856 for the chair of natural philosophy at Marischal College, Aberdeen. He was also interested in the Lowndean Chair of Geometry and Astronomy at Cambridge in 1858 and the chair of astronomy at Glasgow University in the following year. However, he failed to obtain both these and other positions. Perhaps one reason was that, despite a remarkable research record, he had little experience of teaching.

In 1863 Cayley was appointed Sadleirian professor of Pure Mathematics at Cambridge. He was successful despite there being other high quality applicants such as Isaac Todhunter, Norman Ferrers, and Edward Routh. This appointment involved a very large decrease in income for Cayley who now had to manage on a salary only a fraction of that which he had earned as a skilled lawyer. However Cayley was very happy to have the chance to devote himself entirely to mathematics. As Sadleirian professor of Pure Mathematics his duties were:-

... to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science.

Cayley was to more than fulfil these conditions. However, before we look briefly at his remarkable research achievements, we note that he married in 1863, the year of his appointment as Sadleirian professor. On 8 September he married Susan Moline (1831-1923) from Greenwich, the daughter of Robert Moline. Arthur and Susan Cayley had two children: Henry Cayley (1870-1949) who studied mathematics at Cambridge but decided that he couldn't live up to his father's achievements so became an architect, and Mary Cayley (1872-1950). In 1864 Cayley purchased Garden House, Little St Mary's Lane, in Cambridge. On 8 November 1863 Cayley gave his inaugural lecture Analytical geometry - Introductory lecture. However, his lectures over the following years were often based on his latest researches and students did not find them relevant to the Tripos examination in which they all aimed at getting the greatest possible marks [10]:-

Cayley was concerned with mathematics 'per se', while most students wanted competence in a circumscribed syllabus. Any deviation from it was regarded as a waste of time and effort.

He published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work is in developing the algebra of matrices, work on non-euclidean geometry and $n$-dimensional geometry. Of his work on matrices, Richard Feldmann writes [17]:-

Although the term "matrix" was introduced into mathematical literature by James Joseph Sylvester in 1850, the credit for founding the theory of matrices must be given to Arthur Cayley, since he published the first expository articles on the subject. ... Cayley's introductory paper in matrix theory was written in French and published in a German periodical [in 1855]. ... [He] introduces, although quite sketchily, the ideas of inverse matrix and of matrix multiplication, or "compounding" as Cayley called it. The above basic properties are expanded in a second expository article [of 1858] which also lists many additional properties of matrices.

As early as 1849 Cayley had written a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were permutation groups and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups. Here is a part of his definition of an abstract group:-

... the symbol 1 will naturally denote an operation which (either generally or in regard to the particular operand) leaves the operand unaltered. .... A symbol xy denotes the compound operation, the performance of which is equivalent to the performance, first of the operation y, and then of the operation x; xy is of course in general different from yx. But the symbols x, y, ... are in general such that x(yz) = (xy)z etc, so that xyz, xyzt, etc. have a definite signification independent of the particular mode of compounding the symbols ...

Cayley developed the theory of algebraic invariance, and his development of $n$-dimensional geometry has been applied in physics to the study of the space-time continuum. His work on matrices served as a foundation for quantum mechanics, which was developed by Werner Heisenberg in 1925. Cayley also suggested that euclidean and non-euclidean geometry are special types of geometry. He united projective geometry and metrical geometry which is dependent on sizes of angles and lengths of lines.

You can see reviews of Cayley's papers by G B Halsted at THIS LINK.

He only published one book, namely An Elementary Treatise on Elliptic Functions (1876). However, he did contribute Chapter 6 to Peter Guthrie Tait's An Elementary Treatise on Quaternions (1890) and published the six-penny booklet, The Principles of Book-Keeping by Double Entry (1894).

In 1881 he was invited to give a course of lectures at Johns Hopkins University in the United States, where his friend Sylvester was professor of mathematics. He spent January to May of 1882 at Johns Hopkins University where he lectured on Abelian and Theta Functions. After he returned to England, the Royal Society awarded him their Copley Medal. In 1883 Cayley became President of the British Association for the Advancement of Science. In his presidential address Cayley gave an elementary account of his own views of mathematics. His views of geometry were:-

It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration: and that Lobachevsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry. My own view is that Euclid's twelfth axiom in Playfair's form of it does not need demonstration, but is part of our experience - the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann's view ... is that, having 'in intellectu' a more general notion of space (in fact a notion of non-Euclidean space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidean space. But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.

Two descriptions of Cayley, both of him as an old man, are interesting. Macfarlane [15] says:-

... I attended a meeting of the Mathematical Society of London. The room was small, and some twelve mathematicians were assembled round a table, among them was Prof Cayley ... At the close of the meeting Cayley gave me a cordial handshake and referred in the kindest terms to my papers which he had read. He was then about 60 years old, considerably bent, and not filling his clothes. What was most remarkable about him was the active glance of his grey eyes and his peculiar boyish smile.

Thomas Hirst, one of his friends, wrote:-

... a thin weak-looking individual with a large head and face marked with small-pox: he speaks with difficulty and stutters slightly. He never sits upright on his chair but with his posterior on the very edge he leans one elbow on the seat of the chair and throws the other arm over the back.

Here is G B Halsted's tribute [23]:-

Cayley, in addition to his wondrous originality, was assuredly the most learned and erudite of mathematicians. Of him in his science it might be said he knew everything, and he was the very last man who ever will know everything. His was a very gentle, sweet character. Sylvester told me he never saw him angry but once, and that was (both were practicing law!) when a messenger broke in on one of their interviews with a mass of legal documents - new business for Cayley. In an access of disgust, Cayley dashed the documents upon the floor.

We end by giving this tribute from [8]:-

Cayley was Britain's outstanding pure mathematician of the nineteenth century. An algebraist, analyst, and geometer, he was able to link these vast domains of study. More than fifty concepts and theorems of mathematics bear his name. By the end of his life he was revered by mathematicians the world over.

In addition to the Copley Medal we mentioned above, we note that Cayley was elected a fellow of the Royal Society in 1852 and received their Royal Medal in 1859. The London Mathematical Society awarded him their De Morgan Medal in 1884. He was president of the Society 1868-1870. He was elected a fellow of the Royal Society of Edinburgh in 1865. He was elected a fellow of the Royal Astronomical Society in 1857 and served as editor of the Society's publications from 1859 to 1881. In 1872 he was made an honorary fellow of Trinity College, Cambridge. He received honorary degrees from the universities of Cambridge, Oxford, Edinburgh, Dublin, Göttingen, Heidelberg, Leyden and Bologna. A crater on the moon is named for Cayley. He was an honorary foreign member of the French Institute and was elected a fellow of the academies of Berlin, Göttingen, St Petersburg, Milan, Rome, Leyden, Upsala, and Hungary.

After a long period of suffering he died at his home, Garden House, Cambridge. He was buried on 2 February 1895 in Mill Road cemetery, Cambridge. Many representatives from foreign countries attended his funeral in addition to many of the leading British scholars.