威廉·哈密顿William Rowan Hamilton

威廉·哈密顿 的父亲 Archibald Hamilton 没有时间教 哈密顿，因为他经常在外出英格兰处理法律事务。Archibald Hamilton 没有受过大学教育，人们认为 哈密顿 的天赋来自他的母亲萨拉·赫顿。到五岁时，哈密顿 已经学会了拉丁语、希腊语和希伯来语。这些科目是由他的叔叔 James Hamilton 牧师教他的，哈密顿 曾与他在特里姆一起生活多年。詹姆斯是一位出色的教师。

哈密顿 很快掌握了更多语言，8岁时他遇到了美国人泽拉·科尔本。科尔本能够表演惊人的心算绝技，哈密顿 与他一起参加算术能力比赛。[31] 看来输给科尔本激发了 哈密顿 对数学的兴趣。

哈密顿在13岁时开始接触数学，当时他学习了亚历克西斯·克劳德·克莱罗的Algebra，这项任务相对容易一些，因为哈密顿此时已能流利使用法语。15岁时，他开始研究艾萨克·牛顿和皮埃尔·西蒙·拉普拉斯的著作。1822年，哈密顿在皮埃尔·西蒙·拉普拉斯的Mécanique céleste中发现了一个错误，因此引起了爱尔兰皇家天文学家约翰·布林克利的注意，后者说：-

这个年轻人，我不说他将会，而是说他就是，他那个时代的第一位数学家。

哈密顿18岁时进入都柏林三一学院，在第一年他获得了古典学'optime'，这一荣誉每20年才授予一次。

1824年8月，詹姆斯叔叔带 哈密顿 去萨默希尔会见迪士尼一家。正是在这时，哈密顿 第一次见到他们的女儿凯瑟琳，并立刻无可救药地爱上了她。不幸的是，由于他在三一学院还有三年，哈密顿 无法求婚。然而，哈密顿 作为一名本科生取得了显著进步，并在1824年底之前向 爱尔兰皇家科学院 提交了他的第一篇论文，题为 On Caustics。

次年二月，凯瑟琳的母亲告诉 哈密顿，她的女儿将嫁给一位牧师，此人比她年长十五岁。他很富有，能给凯瑟琳提供比 哈密顿 更多的东西。在接下来的一组考试中，哈密顿 因失去凯瑟琳而心烦意乱，只得到了“bene”，而不是通常的“valde bene”。他病倒了，甚至一度考虑自杀。在这一时期，他转向诗歌，这是他在痛苦时期终生保持的习惯。

1826年，哈密顿在科学和古典学两方面都获得了“optime”，这是闻所未闻的，而在他作为本科生的最后一年，他向爱尔兰皇家科学院提交了一篇论文Theory of Systems of Rays。正是在这篇论文中，哈密顿引入了光学的特征函数。

哈密顿的学位考试考官Boyton劝他申请Dunsink天文台皇家天文学家的职位，尽管已经有六位申请人，其中一位是乔治·比德尔·艾里。后来在1827年，董事会任命哈密顿为三一学院Andrews天文学教授，当时他还是年仅二十一岁的本科生。该教授职位带有Royal Astronomer of Ireland的荣誉称号以及居住在Dunsink天文台的待遇。这项任命引起了很大争议，因为哈密顿在观测方面经验不多。他的前任约翰·布林克利教授已成为主教，他认为哈密顿接受这个职位不是正确的决定，并暗示他等待一个会士职位会更明智。结果哈密顿做出了糟糕的选择，因为他对天文学失去了兴趣，把所有时间都花在数学上。

在开始担任这一著名职位之前，哈密顿游历了英格兰和苏格兰（哈密顿家族的发源地）。他遇到了诗人Wordsworth，他们成为了朋友。哈密顿的一个姐妹Eliza也写诗，当Wordsworth来Dunsink拜访时，他喜欢的是她的诗而不是哈密顿的。两人就科学与诗歌进行了长时间的辩论。哈密顿喜欢将两者进行比较，认为数学语言与诗歌一样具有艺术性。然而，Wordsworth不同意，说[4]：-

科学只应用于生活中的物质用途，与想象力作战并希望将其消灭。

Wordsworth不得不相当有力地告诉哈密顿，他的才能在于科学而非诗歌：-

你寄给我大量的诗篇，我非常愉快地接受……然而我们担心这种活动可能会诱使你偏离科学的道路。……我再次冒昧地提请你的考虑，你天性中的诗歌部分是否会在散文领域找到一个更适合其本性的领域，不是因为那些领域更卑微，而是因为它们可以被优雅而有益地踏足，步伐不那么谨慎，韵律不那么精细。

哈密顿收了一个名叫Adare的学生。他们互相产生了不良影响，因为Adare的视力开始出现问题，因为他做了太多观测，同时哈密顿因过度工作而生病。他们决定以度假的方式去Armagh旅行，并拜访另一位天文学家Romney Robinson。正是在这次旅行中，哈密顿遇到了Campbell夫人，她后来成为他最喜爱的知己之一。哈密顿也借此机会拜访了Catherine，因为她住在相对较近的地方，她随后也回访了天文台。哈密顿在她面前如此紧张，以至于在试图向她演示时打碎了望远镜的目镜。这一插曲引发了另一段痛苦和写诗的时期。

1830年7月，哈密顿和他的妹妹伊丽莎访问了华兹华斯，大约就在这个时候，他开始认真考虑结婚。他考虑过艾伦·德维尔，并告诉华兹华斯他[4]：-

……钦佩她的才智……

但他没有提到爱情。然而，他确实用诗歌对她进行了狂轰滥炸，并正要向她求婚时，她碰巧说[5]她

……除了在库拉，在任何地方都无法幸福生活。

哈密顿认为这是她委婉拒绝他的方式，于是他不再追求她。然而，事实证明他错了，因为她第二年就结婚了，并且确实离开了库拉！幸运的是，这件事带来了一件好事，哈密顿与艾伦的兄弟奥布里成了坚定的朋友，尽管1851年关于宗教的争论使他们分道扬镳。

撇开凯瑟琳不谈，哈密顿在与女性的关系上似乎相当反复无常。也许这是因为他认为自己应该结婚，所以如果不能得到凯瑟琳，那么娶谁其实并不重要。最终他娶了海伦·玛丽亚·贝利，她就住在天文台对面的田野那边。哈密顿告诉奥布里她是"not at all brilliant"，不幸的是，这桩婚姻从一开始就注定失败。他们在贝利农场度蜜月，哈密顿在此期间撰写了他Theory of Systems of Rays的第三篇补编。然后在天文台，海伦不太懂持家，又经常生病，家务变得极其混乱。在随后的岁月里，她大部分时间都不在天文台，要么在照顾生病的母亲，要么自己身体不适。

1832年，哈密顿出版了他的Theory of Systems of Rays第三篇补编，这本质上是一篇关于应用于光学的特征函数的论文。在作品接近尾声时，他将特征函数应用于研究奥古斯丁·菲涅耳的波面。由此他预测了锥形折射，并请三一学院物理学教授汉弗莱·劳埃德尝试通过实验验证他的理论预测。汉弗莱·劳埃德在两个月后做到了，这一理论预测给哈密顿带来了巨大声誉。然而，这也引发了与詹姆斯·麦古拉的争议，后者自己已经非常接近这一理论发现，但被迫承认未能迈出最后一步。

1833年11月4日，哈密顿向爱尔兰皇家科学院宣读了一篇论文，将复数表示为代数对，或实数的有序对。他在1834年的On a General Method in Dynamics中运用代数处理动力学。在这篇论文中，哈密顿首次陈述了应用于动力学的特征函数，并在第二年就此主题写了第二篇论文。汉金斯在[1]中写道：-

这些论文读起来很困难。哈密顿像往常一样极其简洁地阐述了他的论证，他的方法与现在教科书里描述该方法时通常呈现的方式完全不同。在两篇关于动力学的论文中，哈密顿首次将特征函数$V$应用于动力学，正如他在光学中所做的那样，特征函数是系统在构型空间中从初始点到终点运动的作用量。通过他的变动作用定律，他把初始坐标和最终坐标作为特征函数的自变量。对于保守系统，总能量$H$沿任何真实路径都是常数，但如果初始点和终点改变，它就会改变，因此动力学中的特征函数成为初始位置和最终位置的6n个坐标（对于$n$个粒子）以及哈密顿量$H$的函数。

1834年，哈密顿和海伦有了一个儿子，哈密顿·埃德温。海伦随后离开邓辛克九个月，留下哈密顿，他更加投入工作以对抗孤独。1835年，哈密顿发表了Algebra as the Science of Pure Time，这些论文受他研究康德启发，并提交给英国协会 for the Advancement of Science的一次会议。这第二篇关于代数偶的论文将代数偶与时间步长等同起来，他把这些偶称为“时间步长”。

哈密顿于1835年被封为爵士，那年他的次子阿奇博尔德·亨利出生，但接下来的几年并未给他带来多少幸福。在发现代数对之后，他试图将理论扩展到三元组，这成为困扰他多年的执念。次年秋天，他去布里斯托参加英国协会的一次会议，海伦带着孩子去了贝利农场十个月。他的表兄阿瑟去世了，不久后海伦从母亲那里回来后，又再次去了英格兰，这次在生下女儿海伦·伊丽莎·阿米莉亚后将孩子们留下。此时，哈密顿变得抑郁，并开始酗酒，于是他的姐妹回到邓辛克居住。

海伦在1842年回来时，哈密顿如此专注于三元组，甚至连他的孩子们都察觉到了。每天早上他们都会问[26]：-

那么，爸爸，你能乘三元组吗？

但他不得不承认，他仍然只能对它们进行加法和减法。

1843年10月16日（星期一），哈密顿和妻子沿着皇家运河散步，去主持爱尔兰皇家科学院的一次理事会会议。尽管妻子不时和他说话，哈密顿几乎没听见，因为四元数——第一个被研究的非交换代数——的发现正在他脑海中成形：-

在这里，我萌生了一个想法：为了用三元组进行计算，我们必须在某种意义上承认空间的第四维……一个电路似乎闭合了，火花闪现。

他无法抗拒冲动，要把四元数的公式刻在

$i^{2} = j^{2} = k^{2} = i j k = -1$。

布鲁姆桥（他称之为布罗汉姆桥）的石头上，当时他和妻子经过那里。

1958年，Royal Irish Academy立了一块牌匾来纪念此事。见THIS LINK。
据说他正在雕刻它的版画在THIS LINK。

哈密顿觉得这一发现将彻底改变数学物理，他余生都在研究四元数。他写道[26]：-

我仍然必须断言，这一发现在我看来对十九世纪中叶的重要性，正如流数[微积分]的发现对十七世纪末的重要性一样。

哈密顿发现四元数后不久，他的个人生活又开始困扰他。1845年，Thomas Disney在天文台拜访了哈密顿，并带上了Catherine。这一定让哈密顿感到不安，因为他的酒精依赖恶化了。在次年二月的Macfarlane17会议上，他因醉酒而出丑。Macfarlane[17]写道：-

……在都柏林一个科学团体的晚宴上，他失去了自制力，事后深感羞辱，于是在朋友们的建议下决心彻底戒酒。这个决心他保持了两年，那时……他因坚持喝水而受到嘲笑，尤其是被乔治·比德尔·艾里嘲笑……。他打破了自己的好决心，从那时起，对酒精刺激物的渴望就一直缠着他。

1847年，他的叔叔詹姆斯和威利去世，他在三一学院的同事詹姆斯·麦古拉自杀，这使他深受困扰，尽管他们并不总是意见一致。第二年，凯瑟琳开始给哈密顿写信，这在这段抑郁时期不可能没有影响。通信持续了六周，变得越来越随意和私人化，直到凯瑟琳感到如此内疚，以至于向丈夫坦白。哈密顿写信给巴洛，告知他他们再也不会收到他的信了。然而，凯瑟琳又写了一次信，这次她因悔恨至极而（未遂地）试图自杀。此后她余生都与母亲或兄弟姐妹住在一起，尽管与巴洛没有正式分居。哈密顿坚持给凯瑟琳写信，通过她的亲戚转交。

毫不奇怪，哈密顿在此之后立即屈服于酒精，但他投身于工作，开始写他的Lectures on Quaternions。他于1853年出版了Lectures on Quaternions，但他很快意识到这不是一本学习四元数理论的好书。也许哈密顿作为教师的缺乏技巧在这部作品中显现出来。

哈密顿帮助凯瑟琳的儿子詹姆斯准备他的会士考试，考试内容是关于四元数的。他认为这是对巴洛的报复，因为他能够以他父亲无法做到的方式帮助他的儿子。那年晚些时候，哈密顿收到凯瑟琳送的一个铅笔盒，上面刻着[5]：-

来自一个你永远不应忘记、也不应怀有恶意的人，如果我们能再次相见，她会死得更满足。

哈密顿径直去找凯瑟琳，给了她一本Lectures on Quaternions。她两周后去世了。作为处理悲痛的一种方式，哈密顿用不断的通信纠缠迪士尼一家，有时一天写两封信。坎贝尔夫人是另一个承受邮件负担的人，因为只有她和迪士尼一家知道他对凯瑟琳的爱。另一方面，海伦一定一直怀疑自己并未在丈夫心中占据首位，这种想法在1855年她发现多拉·迪士尼（凯瑟琳的嫂子）的一封信时一定得到了加强。这引发了一场争吵，尽管唯一的后果是多拉让她的丈夫代写地址，他们并没有完全停止通信。

哈密顿 决心写出一部具有持久价值的著作，于是开始写另一本书 Elements of Quaternions，他估计这本书将有400页，需要2年时间写成。书名表明 哈密顿 以 欧几里得 的 Elements 为范本，事实也确实如此。这本书最终长度是原计划的两倍，花了七年时间写成。事实上，他去世时最后一章尚未完成，这本书最终出版时附有他儿子 哈密顿 Edwin Hamilton 写的序言。

并非每个人都发现哈密顿的四元数能解答他们一直在寻找的一切。开尔文写道：-

四元数来自哈密顿在完成他真正出色的工作之后，尽管它巧妙绝伦，却给任何接触过它的人带来了纯粹的祸害。

阿瑟·凯莱将四元数比作一张袖珍地图[26]：-

……它包含了一切，但必须展开成另一种形式才能被理解。

哈密顿在收到他当选为美国国家科学院首位外籍院士的消息后不久，因严重痛风发作而去世。

William Rowan Hamilton's father, Archibald Hamilton, did not have time to teach William as he was often away in England pursuing legal business. Archibald Hamilton had not had a university education and it is thought that Hamilton's genius came from his mother, Sarah Hutton. By the age of five, William had already learned Latin, Greek, and Hebrew. He was taught these subjects by his uncle, the Rev James Hamilton, who William lived with in Trim for many years. James was a fine teacher.

William soon mastered additional languages and at the age of 8 he met the American Zerah Colburn. Colburn could perform amazing mental arithmetical feats and Hamilton joined in competitions of arithmetical ability with him. [31] It appears that losing to Colburn sparked Hamilton's interest in mathematics.

Hamilton's introduction to mathematics came at the age of 13 when he studied Clairaut's Algebra, a task made somewhat easier as Hamilton was fluent in French by this time. At age 15 he started studying the works of Newton and Laplace. In 1822 Hamilton found an error in Laplace's Mécanique céleste and, as a result of this, he came to the attention of John Brinkley, the Royal Astronomer of Ireland, who said:-

This young man, I do not say will be, but is, the first mathematician of his age.

Hamilton entered Trinity College, Dublin at the age of 18 and in his first year he obtained an 'optime' in Classics, a distinction only awarded once in 20 years.

In August 1824, Uncle James took Hamilton to Summerhill to meet the Disney family. It was at this point that William first met their daughter Catherine and immediately fell hopelessly in love with her. Unfortunately, as he had three years left at Trinity College, Hamilton was not in a position to propose marriage. However Hamilton was making remarkable progress for an undergraduate and submitted his first paper to the Royal Irish Academy before the end of 1824, which was entitled On Caustics.

The following February, Catherine's mother informed William that her daughter was to marry a clergyman, who was fifteen years her senior. He was affluent and could offer more to Catherine than Hamilton. In his next set of exams William was given a 'bene' instead of the usual 'valde bene' due to the fact that he was so distraught at losing Catherine. He became ill and at one point he even considered suicide. In this period he turned to poetry, which was a habit that he pursued for the rest of his life in times of anguish.

In 1826 Hamilton received an 'optime' in both science and Classics, which was unheard of, while in his final year as an undergraduate he presented a memoir Theory of Systems of Rays to the Royal Irish Academy. It is in this paper that Hamilton introduced the characteristic function for optics.

Hamilton's finals examiner, Boyton, persuaded him to apply for the post of Royal Astronomer at Dunsink observatory even although there had already been six applicants, one of whom was George Biddell Airy. Later in 1827 the board appointed Hamilton Andrews' Professor of Astronomy in Trinity College while he was still an undergraduate aged twenty-one years. The professorship carried the honorary title Royal Astronomer of Ireland and the benefit of residing at Dunsink Observatory. This appointment brought a great deal of controversy as Hamilton did not have much experience in observing. His predecessor, Professor Brinkley, who had become a bishop, did not think that it had been the correct decision for Hamilton to accept the post and implied that it would have been prudent for him to have waited for a fellowship. It turned out that Hamilton had made an poor choice as he lost interest in astronomy and spend all time on mathematics.

Before beginning his duties in this prestigious position, Hamilton toured England and Scotland (from where the Hamilton family originated). He met the poet Wordsworth and they became friends. One of Hamilton's sisters Eliza wrote poetry too and when Wordsworth came to Dunsink to visit, it was her poems that he liked rather than Hamilton's. The two men had long debates over science versus poetry. Hamilton liked to compare the two, suggesting that mathematical language was as artistic as poetry. However, Wordsworth disagreed saying that [4]:-

Science applied only to material uses of life waged war with and wished to extinguish imagination.

Wordsworth had to tell Hamilton quite forcibly that his talents were in science rather than poetry:-

You send me showers of verses which I receive with much pleasure ... yet have we fears that this employment may seduce you from the path of science. ... Again I do venture to submit to your consideration, whether the poetical parts of your nature would not find a field more favourable to their nature in the regions of prose, not because those regions are humbler, but because they may be gracefully and profitably trod, with footsteps less careful and in measures less elaborate.

Hamilton took on a pupil by the name of Adare. They were a bad influence on each other as Adare's eyesight started to present problems as he was doing too much observing, while at the same time Hamilton became ill due to overwork. They decided to take a trip to Armagh by way of a holiday and visit another astronomer Romney Robinson. It was on this occasion that Hamilton met Lady Campbell, who was to become one of his favourite confidants. William also took the opportunity to visit Catherine, as she was living relatively nearby, which she then reciprocated by coming to the observatory. Hamilton was so nervous in her presence that he broke the eyepiece of the telescope whilst trying to give her a demonstration. This episode inspired another interval of misery and poem writing.

In July 1830 Hamilton and his sister Eliza visited Wordsworth and it was around this time that he started to think seriously about getting married. He considered Ellen de Vere, and he told Wordsworth that he [4]:-

... admired her mind ...

but he did not mention love. He did, however, bombard her with poetry and was about to propose marriage when she happened to say [5] that she could

... not live happily anywhere but at Curragh.

Hamilton thought this was her way of discouraging him tactfully and so he ceased to pursue her. However he was proved to be mistaken as she married the following year and did leave Curragh! Fortunately, one good thing transpired from the event as Hamilton became firm friends with Ellen's brother Aubrey although a dispute about religion in 1851 made them go their separate ways.

Catherine aside, Hamilton seemed quite fickle when it came to relationships with women. Perhaps this was because he thought that he ought to marry and so, if he could not have Catherine, then it did not really matter whom he married. In the end he married Helen Maria Bayly who lived just across the fields from the observatory. William told Aubrey that she was "not at all brilliant" and, unfortunately, the marriage was fated from the start. They spent their honeymoon at Bayly Farm and Hamilton worked on his third supplement to his Theory of Systems of Rays for the duration. Then at the observatory Helen did not have much of an idea of housekeeping and was so often ill that the household became extremely disorganised. In the years to come she spent most of her time away from the observatory as she was looking after her ailing mother or was indisposed herself.

In 1832 Hamilton published this third supplement to Theory of Systems of Rays which is essentially a treatise on the characteristic function applied to optics. Near the end of the work he applied the characteristic function to study Fresnel's wave surface. From this he predicted conical refraction and asked the Professor of Physics at Trinity College, Humphrey Lloyd, to try to verify his theoretical prediction experimentally. This Lloyd did two months later and this theoretical prediction brought great fame to Hamilton. However, it also led to controversy with MacCullagh, who had come very close to the theoretical discovery himself but, he was forced to admit, had failed to take the last step.

On 4 November 1833 Hamilton read a paper to the Royal Irish Academy expressing complex numbers as algebraic couples, or ordered pairs of real numbers. He used algebra in treating dynamics in On a General Method in Dynamics in 1834. In this paper Hamilton gave his first statement of the characteristic function applied to dynamics and wrote a second paper on the topic the following year. Hankins writes in [1]:-

These papers are difficult to read. Hamilton presented his arguments with great economy, as usual, and his approach was entirely different from that now commonly presented in textbooks describing the method. In the two essays on dynamics Hamilton first applied the characteristic function $V$ to dynamics just as he had in optics, the characteristic function being the action of the system in moving from its initial to its final point in configuration space. By his law of varying action he made the initial and final coordinates the independent variables of the characteristic function. For conservative systems, the total energy $H$ was constant along any real path but varied if the initial and final points were varied, and so the characteristic function in dynamics became a function of the 6n coordinates of initial and final position (for $n$ particles) and the Hamiltonian $H$.

The year 1834 was the one in which Hamilton and Helen had a son, William Edwin. Helen then left Dunsink for nine months leaving Hamilton to fight the loneliness by throwing himself into his work even more. In 1835 Hamilton published Algebra as the Science of Pure Time which were inspired by his study of Kant and presented to a meeting of the British Association for the Advancement of Science. This second paper on algebraic couples identified them with steps in time and he referred to the couples as 'time steps'.

Hamilton was knighted in 1835 and that year his second son, Archibald Henry, was born but the next few years did not bring him much happiness. After the discovery of algebraic couples, he tried to extend the theory to triplets, and this became an obsession that plagued him for many years. The following autumn he went to Bristol for a meeting of the British Association, and Helen took the children with her to Bayly Farm for ten months. His cousin Arthur died, and not long after Helen returned from her mother's she went away again to England this time leaving the children behind after the birth of a daughter, Helen Eliza Amelia. At this point, William became depressed and started to have problems with alcohol so his sister came back to live at Dunsink.

Helen returned in 1842 when Hamilton was so preoccupied with the triplets that even his children were aware of it. Every morning they would inquire [26]:-

Well, Papa can you multiply triplets?

but he had to admit that he could still only add and subtract them.

On 16 October 1843 (a Monday) Hamilton was walking in along the Royal Canal with his wife to preside at a Council meeting of the Royal Irish Academy. Although his wife talked to him now and again Hamilton hardly heard, for the discovery of the quaternions, the first noncommutative algebra to be studied, was taking shape in his mind:-

And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.

He could not resist the impulse to carve the formulae for the quaternions

$i^{2} = j^{2} = k^{2} = i j k = -1$.

in the stone of Broome Bridge (or Brougham Bridge as he called it) as he and his wife passed it.

In 1958 the Royal Irish Academy erected a plaque commemorating this. See THIS LINK.
An engraving of him supposedly carving it is at THIS LINK.

Hamilton felt this discovery would revolutionise mathematical physics and he spent the rest of his life working on quaternions. He wrote [26]:-

I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions [the calculus] was for the close of the seventeenth.

Shortly after Hamilton's discovery of the quaternions his personal life started to prey on his mind again. In 1845, Thomas Disney visited Hamilton at the observatory and brought Catherine with him. This must have upset William as his alcohol dependency took a turn for the worse. At a meeting of the Geological Society the following February he made an exhibition of himself through his intoxication. Macfarlane [17] writes:-

... at a dinner of a scientific society in Dublin he lost control of himself, and was so mortified that, on the advice of friends he resolved to abstain totally. This resolution he kept for two years, when ... he was taunted for sticking to water, particularly by Airy ... . He broke his good resolution, and from that time forward the craving for alcoholic stimulants clung to him.

The year 1847 brought the deaths of his uncles James and Willey and the suicide of his colleague at Trinity College, James MacCullagh, which greatly disturbed him despite the fact that they had not always seen eye to eye. The following year Catherine began writing to Hamilton, which cannot have helped at this time of depression. The correspondence continued for six weeks and became more informal and personal until Catherine felt so guilty that she confessed to her husband. Hamilton wrote to Barlow and informed him that they would never hear from him again. However, Catherine wrote once more and this time attempted suicide (unsuccessfully) as her remorse was so great. She then spent the rest of her life living with her mother or siblings, although there was no official separation from Barlow. Hamilton persisted in his correspondence to Catherine, which he sent through her relatives.

It is no surprise that Hamilton gave in to alcohol immediately after this, but he threw himself into his work and began writing his Lectures on Quaternions. He published Lectures on Quaternions in 1853 but he soon realised that it was not a good book from which to learn the theory of quaternions. Perhaps Hamilton's lack of skill as a teacher showed up in this work.

Hamilton helped Catherine's son James to prepare for his Fellowship examinations which were on quaternions. He saw this as revenge towards Barlow as he was able to help his son in a way that his father could not. Later that year Hamilton received a pencil case from Catherine with an inscription that read [5]:-

From one who you must never forget, nor think unkindly of, and who would have died more contented if we had once more met.

Hamilton went straight to Catherine and gave her a copy of Lectures on Quaternions. She died two weeks later. As a way of dealing with his grief, Hamilton plagued the Disney family with incessant correspondence, sometimes writing two letters a day. Lady Campbell was another sufferer of the burden of mail, as only she and the Disneys knew of his love for Catherine. On the other hand, Helen must have always suspected that she did not take first place in her husband's heart, a notion that must have been strengthened in 1855 when she found a letter from Dora Disney (Catherine's sister-in-law). This led to an argument, although the only consequence was that Dora had her letters addressed by her husband, they did not stop altogether.

Determined to produce a work of lasting quality, Hamilton began to write another book Elements of Quaternions which he estimated would be 400 pages long and take 2 years to write. The title suggests that Hamilton modelled his work on Euclid's Elements and indeed this was the case. The book ended up double its intended length and took seven years to write. In fact the final chapter was incomplete when he died and the book was finally published with a preface by his son William Edwin Hamilton.

Not everyone found Hamilton's quaternions the answer to everything they had been looking for. Thomson wrote:-

Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.

Cayley compared the quaternions with a pocket map [26]:-

... which contained everything but had to be unfolded into another form before it could be understood.

Hamilton died from a severe attack of gout shortly after receiving the news that he had been elected the first foreign member of the National Academy of Sciences of the USA.