彼得·格思里·泰特Peter Guthrie Tait

彼得·格思里·泰特的父亲是John Tait，母亲是Mary Ronaldson。John Tait是第五代巴克卢公爵Walter Francis Scott的秘书。泰特有两个姐妹，他在达尔基斯的文法学校开始上学。然而，他六岁时父亲去世，泰特与两个姐妹和母亲搬到爱丁堡，与叔叔弗瑞兹·约翰 Ronaldson同住。作为爱丁堡的银行家，约翰 Ronaldson却对科学感兴趣，尤其是天文学、地质学以及新发明的摄影术。他很快让年轻的侄子泰特对这些学科产生了兴趣，可以说泰特对科学的兴趣直接源于他叔叔对科学的热情。

当全家搬到爱丁堡时，泰特当然不得不离开他在达尔基斯的学校。他接下来就读于爱丁堡Circus Place的一所私立学校，然后于1841年，当他十岁时，进入了爱丁堡学院。Lewis Campbell后来成为圣安德鲁斯大学的希腊语教授，詹姆斯·克拉克·麦克斯韦在学院比泰特高一年级。事实上詹姆斯·克拉克·麦克斯韦比泰特稍年轻，所以一年的差距当然并不反映他们各自的年龄。

泰特在爱丁堡学院的六年中每年都是班级第一名。然而，他早期的兴趣不在科学，而在古典学。到学院第四年时，数学已成为他真正的热爱，也是他真正擅长的科目。1846年，他在爱丁堡学院俱乐部奖的数学部分中名列第一，这绝非易事，因为他击败了名列第二的Lewis Campbell和名列第三的詹姆斯·克拉克·麦克斯韦。1847年，泰特在爱丁堡学院的最后一年，詹姆斯·克拉克·麦克斯韦报了仇，他在爱丁堡学院俱乐部奖中名列第一，泰特第二。

1847年11月，16岁的泰特进入爱丁堡大学。詹姆斯·克拉克·麦克斯韦同时以泰特身份进入爱丁堡大学，他们一起参加了由菲利普·凯兰教授的第二数学班和由James David Forbes教授的自然哲学（物理学）班。泰特只在爱丁堡大学待了一年，然后于1848年进入剑桥大学彼得豪斯学院。在那里，他由威廉·霍普金斯辅导，度过了一段非凡的本科生涯。1852年1月，二十岁的他以数学荣誉学位考试高级数学荣誉学位考试一等及格者（Wrangler）毕业。这意味着他在当年剑桥大学授予的一等数学学位中名列第一。他还是第一位史密斯奖得主。

詹姆斯·克拉克·麦克斯韦于1850年跟随泰特进入彼得豪斯学院，但转到了三一学院，他认为在那里更容易获得 fellowship。泰特的另一位同学和朋友是William Steele，他与泰特同年，以Second Wrangler毕业。泰特在彼得豪斯学院获得了Fellowship，除了辅导本科生参加荣誉学位考试外，他开始与凯瑟琳·斯蒂尔合作撰写Dynamics of a particle。不幸的是，斯蒂尔在书写成之前就去世了，但泰特继续了这个项目，并慷慨地以他们的联合署名出版了这本书，尽管大部分是他自己写的。该书于1856年出版。

1854年9月，泰特接受了贝尔法斯特女王学院数学教授的职位。他在贝尔法斯特结识的许多同事和朋友对泰特的职业生涯产生了非常重大的影响。其中一位是Thomas Andrews，两人合作进行实验，以确定臭氧的密度，以及使放电通过氧气和其他气体的影响。泰特在此之前没有参与过实验工作，正是由于Andrews的影响，他才将这一兴趣加入到他不断增长的技能范围中。与Andrews一起进行的这项研究将泰特引向化学，这是他整个职业生涯中一直保持兴趣的学科。

另一个真正重要的友谊是与威廉·哈密顿的友谊。泰特在1853年还在剑桥时就读过威廉·哈密顿的Lectures on 四元数，但尽管这个主题吸引了他，当时他更专注于数学的物理应用，在那个阶段并没有深入这个主题。然后在1858年7月，泰特读了赫尔曼·冯·亥姆霍兹在Crelle's Journal上发表的一篇关于理想流体运动的论文。赫尔曼·冯·亥姆霍兹的论文Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechenⓉ（论与涡旋运动相对应的流体动力学方程的积分）一开始就把理想流体的运动分解为平移、旋转和变形。泰特看到，使用四元数他可以把流体速度表示为一个“向量函数”。正是威廉·哈密顿的四元数微分学随后给出的物理洞察力给泰特留下了深刻印象，他开始努力发展一种物理理论。

泰特于1858年8月开始与威廉·哈密顿通信，在回复威廉·哈密顿关于他如何开始研究四元数的问题时，泰特于1858年12月7日写信给威廉·哈密顿（例如见[7]）：-

...直到去年八月，我突然想起多年前在您讲义第610页上欣赏过的某些公式——我认为（现在仍然认为）它们可能正好符合我的目的。（更直接让我想到这一点的是赫尔曼·冯·亥姆霍兹在奥古斯都·利奥波德·克雷勒杂志（第LX卷）上的一篇论文，我去年七月一收到就在读...标题（德语）我忘了——但我现在手边有一份我自己的手稿翻译，标题是“涡旋运动”...）。

泰特与威廉·哈密顿的友谊对他未来的研究至关重要，而泰特建立的其他友谊在他的家庭生活中也很重要。他在彼得豪斯学院的两个朋友是詹姆斯·波特牧师的儿子，通过他们，泰特遇到了他们的妹妹玛格丽特·阿彻·波特，他于1857年10月13日在贝尔法斯特与她结婚。

1859年，爱丁堡大学的自然哲学讲席出现空缺，因为J·D·福布斯已转往圣安德鲁斯大学。泰特是该讲席的候选人，但詹姆斯·克拉克·麦克斯韦也是，后者在阿伯丁的马歇尔学院与国王学院合并后被迫另谋职位。爱德华·约翰·劳思也曾是候选人，他在詹姆斯·克拉克·麦克斯韦那年是剑桥的第一名优等生，但真正的竞争始终是在泰特与詹姆斯·克拉克·麦克斯韦之间。尽管詹姆斯·克拉克·麦克斯韦取得了杰出的科学成就，泰特还是胜出了。当爱丁堡的报纸Courant报道这一结果时，它指出泰特之所以被选中而非詹姆斯·克拉克·麦克斯韦，是因为：-

……在我们这样的大学里，教授还应具备另一种品质，那就是在假定学生知识不完善、甚至完全无知的前提下进行口头讲解的能力。

关于泰特更适合教导基础薄弱学生的说法当然是公允的，而且，泰特的个性自然使他在任命委员会那里留下了比更为内敛的詹姆斯·克拉克·麦克斯韦更强烈的印象。

到1860年他抵达爱丁堡时，泰特已在应用威廉·哈密顿的四元数方面做出了重要贡献。就在他出任爱丁堡自然哲学讲席的那一年，他发表了Quaternion investigations connected with electro-dynamics and magnetism，其中他用四元数的语言重新表述了赫尔曼·冯·亥姆霍兹的流体动力学-电磁类比。正如11在[11]中所写：-

……不仅四元数分析因获得一种新的物理意义而受益。四元数公式还有助于更容易地把握可用流体运动来描述的物理情形。

到1863年，当他在Proceedings of the Royal Society of Edinburgh上发表Note on a quaternion transformation时，泰特声称：-

……数学物理的下一次重大扩展，极有可能将由四元数提供。

威廉·哈密顿于1865年去世，泰特接过了这场运动，力图使四元数在数学物理中发挥主导作用。在他对这一主题所作的众多贡献中，我们应当提到他的两部重要著作Elementary Treatise on Quaternions（1867年）和Introduction to Quaternions（1873年）。詹姆斯·克拉克·麦克斯韦对泰特关于四元数物理应用的众多著作印象深刻，并于1871年在给开尔文的一封信中写道：——

你应当让世人知道，适用于物理学的数学方法的真正源头，可以在⟦L1⟧的⟦E1⟧中找到。向量与四元数的体积分、面积分和线积分及其性质，正如泰特正在逐步推导出来的那样，其价值抵得上其他学术重镇正在进行的一切工作。

尽管泰特在四元数上投入了大量精力，他还参与了许多其他活动。1862年，他与James A Wanklyn合作发表了关于蒸发和冒泡过程中产生的电的论文。三年后，他发表了一篇论文，讨论铁屑在置于磁场中的振动板上的运动。1866年，他与物理学家Balfour Stewart开始了一个合作项目，研究在真空中快速旋转的圆盘的加热问题。这是泰特在整个职业生涯中多次回到的一个主题。然后，在1867年，除了关于四元数的论著之外，他还发表了他对赫尔曼·冯·亥姆霍兹1858年文章的翻译，以及也许使他最为人所知的Treatise on Natural Philosophy。

1861年，泰特一直在撰写一部关于数学物理的著作。他的朋友开尔文（后来的开尔文勋爵）：——

……令我大为高兴的是，他主动提出加入。

两人打算写一部两卷本的著作，Treatise on Natural Philosophy（1867）将是其中的第一卷。然而第二卷从未写成。这部被称为T & T'的论著主要由泰特撰写，他似乎尽管有其他众多活动仍能找到时间，而开尔文则发现他的许多其他活动使他无法像泰特那样找到足够的时间来写这本书。Hamilton 伦纳德·尤金·迪克森在[10]中写道：-

这部著作是划时代的，在科学发展中引发了一场革命。T & T'，正如作者们自称的那样，首次将“能量守恒”的概念追溯到艾萨克·牛顿，这一概念当时刚刚在物理学家中获得认可，他们一劳永逸地表明“能量”是基本的物理实体，而它的“守恒”是其首要的、支配一切的性质。

我们已经详细介绍了泰特在1867年的重大成就，但那一年的还有一个事件我们应该提及。赫尔曼·冯·亥姆霍兹在他1858年的论文中描述了涡环的理论行为。他声称两个相互作用的环在相互作用时会改变大小和速度，但会保持其环形。泰特在1867年用烟圈实验验证了赫尔曼·冯·亥姆霍兹的理论主张。他使用了两个各带有一个橡胶膜的盒子，当膜被敲击时，会射出白色的烟圈。开尔文于1867年1月22日写信给赫尔曼·冯·亥姆霍兹：-

……几天前泰特在爱丁堡向我展示了一种产生[涡环]的绝妙方法。我们有时能让一个环穿过另一个环，完美地说明了你的描述；当一个环靠近另一个环时，每个环都受到很大扰动，可以看到它在几秒钟内处于剧烈振动状态，直到它再次稳定成圆形。……这些振动为数学工作提供了一个美丽的主题。

这些实验对开尔文产生了重大影响，他将形式的永久性视为原子的一种可能解释，从而解释了不同元素可以如何构建。泰特起初并不信服开尔文的想法，当然这是正确的，因为尽管这是一个美丽的想法，但它完全错误。这个想法导致泰特、开尔文和詹姆斯·克拉克·麦克斯韦开始研究纽结理论，因为在开尔文的涡原子理论中，基本的构建块将是在三维中打结的环。根据赫尔曼·冯·亥姆霍兹的完美流体理论，这些打结的环，尽管可以被扭曲，但会保持“相同的结”，就像一根圆形打结的绳子可以四处移动，但结的形式保持不变。

泰特、开尔文和詹姆斯·克拉克·麦克斯韦互相通信，在研究纽结时发明了许多拓扑学想法。不久他们发现了利斯廷1847年对纽结理论的贡献。泰特尽管起初不信服开尔文的涡原子理论，但在1870年代初开始将该理论纳入他在爱丁堡的讲座课程，并作了描述该理论的通俗讲座。1876年，泰特开始深入研究纽结，试图对它们进行分类。他在1876-77学年在Proceedings of the Royal Society of Edinburgh上发表了七篇关于纽结的论文。

泰特考虑了交错纽结，即在二维空间中沿投影行进时，交叉点交替上下穿过的纽结。他将这种纽结的$n$个交叉点标记为$A, B, C, ...$，然后该纽结将由长度为2$n$的交叉点序列来描述，其中当纽结被行进时，$A, B, C, ...$中的每一个恰好出现两次。于是有两个基本问题需要解决。第一，上述类型的哪些序列对应于一个纽结；第二，如何确定由这样的序列描述的两个纽结何时是相同的。

在没有任何严格理论的情况下——那将远远超出十九世纪数学的范围——泰特开始利用他的数学和几何直觉对纽结进行分类。到1877年，他已经分类了所有七个交叉点的纽结，但就此停止了。此后他考虑的问题之一是图的着色，因为他声称拥有四色定理的正确证明。他的证明是错误的，遗憾的是，他没有将图的着色与他几年前考虑过的纽结理论联系起来。他多年研究的另一个课题是挑战者号探险关于深海温度的结果。1881年，泰特发表了一篇关于该课题的重要论文，其中他展示了如何因温度计承受高压而校正温度读数。

他在1883年对爱丁堡数学会的演讲中回到了纽结的话题：-

我们发现，画出具有任意给定交叉数的所有可能纽结，仅仅成了一个熟练劳动的问题。所需的劳动量随着交叉数的增加而极其迅速地增长。……除了七个交叉的纽结之外，我没能找到时间将这一过程推进得更远。……非常希望有某位具备必要闲暇的人，能够尝试扩展这份列表，如果可能的话直到11个……

托马斯·柯克曼读了泰特演讲的文本，并开始着手对七个交叉以上的纽结进行分类。1884年5月，他把关于最多九个交叉的纽结投影的结果寄给了泰特，但他还没有考虑判定哪些投影会导致等价纽结的问题。泰特研究了这一问题的另一方面，并且仅考虑交错纽结，在几周内就解决了等价性问题。泰特似乎知道如何在不使用严格方法的情况下判断两个纽结是否等价。他在那篇列出纽结表的论文中相当清楚地说明了这一点，他说他的方法：-

……的缺点是或多或少带有尝试性。并不是说所制定的规则……留有任何仅凭猜测的余地，而是它们过于复杂，无法始终完全保持在视野之中。因此，我们不能绝对确定通过这些过程，我们已经获得了我们所采用的定义所包含的所有本质上不同的形式。

尽管存在这些问题，泰特却完全清楚自己在做什么，因为引人注目的是，他的表是正确的。当托马斯·柯克曼在1885年1月把所有具有10个交叉的纽结投影寄给他时，泰特再次发现它们全都属于等价纽结。这些表于1885年9月印刷，而且它们再次完全正确。到那时，他已经从托马斯·柯克曼那里收到了1581个具有11个交叉的纽结投影，而这一次泰特觉得自己没有时间来解决这些纽结的等价性问题。

在一篇这样长度的文章中，不可能涵盖泰特研究过的所有主题。Knott [7]列出了泰特撰写的365篇论文和22本书。我们将提到他在结束纽结工作后研究的最后两个主题。开尔文建议他研究气体的动理论，在1886年至1892年间，泰特发表了20多篇关于该主题的论文。在这项工作中，他给出了开尔文认为是约翰·詹姆斯·沃特斯顿-詹姆斯·克拉克·麦克斯韦均分定理的第一个证明。

泰特还写了一篇关于高尔夫球轨迹的经典论文（1896年）。高尔夫这个主题是泰特非常感兴趣的。在他的四个儿子中，第三个是弗雷德里克泰特。他在1893年成为领先的业余高尔夫球手，并在1896年和1898年赢得了业余锦标赛。他在1896年和1897年的公开锦标赛中获得第三名，并六次在公开赛中成为领先业余选手。弗雷迪泰特，正如他在高尔夫界所知，在圣安德鲁斯有一条以他命名的街道，离我[EFR]的家不远。弗雷迪是黑卫士的一名军人。当他在1899年自愿参加布尔战争时，他放弃了高尔夫。他于1899年12月19日在马格斯方丹受伤，并于1900年2月7日在库杜斯堡的战斗中阵亡。

泰特是一个虔诚的宗教信徒，他与物理学家Balfour Stewart合写了The Unseen Universe（1875）：-

……以纯粹科学的论证来推翻唯物主义。

由于公众的需求，他写了续篇Paradoxical Philosophy（1878年）。

我们在上面给出的细节中描绘了泰特非常正面的形象。这是正确的，因为他值得如此。然而，他的性格还有另一面，我们应该提到。他与他的科学家同行卷入了许多争论，并至少两次参与了非常公开的争论。泰特在这种情况下容易让情感支配理智，他在科学辩论中往往处于下风。他的争论之一是与奥利弗·赫维赛德和约西亚·威拉德·吉布斯的争论，他长期强烈反对他们的向量方法。当然，泰特在这场争论中处于最不利的地位，也许他的心过于专注于四元数方法，以至于他的头脑无法看到奥利弗·赫维赛德和约西亚·威拉德·吉布斯思想的重要性。

另一场激烈的争论是与鲁道夫·克劳修斯和Tyndall的。泰特的爱国心达到了这样的程度，以至于他会让这种考虑影响他对科学的看法。争论始于谁最先提出功与热的等价性。泰特和Tyndall开始争论是Joule还是Julius Robert Mayer von Mayer拥有优先权。泰特写了一篇极具偏见的热力学史，愚蠢地亲英，Tyndall感到被冒犯是有道理的。后来，当Tyndall请威廉·霍普金斯寄给他所有von Mayer的论文时，威廉·霍普金斯卷入了这场争议，但他在1868年发表的一篇文章中，像泰特亲英一样亲德，声称不仅von Mayer有优先权，德意志民族也有。泰特和鲁道夫·克劳修斯之间更激烈的争论始于1872年，当时詹姆斯·克拉克·麦克斯韦发表了他的Theory of Heat。鲁道夫·克劳修斯指出，英国人在热理论方面试图声称比他们应得的更多，鉴于泰特的写作，这是一个公正的评论。然而，詹姆斯·克拉克·麦克斯韦多年来充分认可了鲁道夫·克劳修斯的贡献，不像泰特那样带有偏见。

当然，泰特的爱国心也意味着他是爱丁堡皇家学会的忠实支持者，从他在爱丁堡被任命为讲席后不久当选为会士起，他就一直忠实地为其服务。他从1879年到1901年担任该学会的秘书长22年。他获得了该学会的Gunning Victoria Jubilee奖，并两次获得Keith奖。虽然从未当选为伦敦皇家学会的会士，但他确实有在1886年获得该学会皇家奖章的殊荣。授予泰特的其他荣誉包括格拉斯哥大学和爱尔兰大学授予的荣誉学位，以及当选为丹麦、荷兰、瑞典和爱尔兰科学院的荣誉会员。

在彼得豪斯举行的一次仪式上，为展示泰特的肖像，开尔文谈到了他的朋友：-

我记得泰特曾说过，除了科学，没有什么值得为之生活。这是真诚地说出的，但泰特自己证明这并不正确。泰特是个博览群书的人。他能把莎士比亚、狄更斯和萨克雷的作品背得滚瓜烂熟。他的记忆力惊人。他一旦带着共鸣读过什么，此后就永远记得。因此，他总是能随时引用令人愉快的段落，这些引用照亮了我们的工作时光。因为我们做的是繁重的数学工作，砸石头根本不算什么。

在[3]中，他被描述如下：-

他熟悉的身影带有某种古怪，或者说不修边幅，他的一些亲密朋友几乎不记得曾见过他穿礼服。外出用餐在他眼里确实令人厌恶，除非是与一两个志同道合的人非正式地聚在一起。

P G Tait's father was John Tait and his mother was Mary Ronaldson. John Tait was a secretary to Walter Francis Scott, the fifth duke of Buccleuch. Peter had two sisters and he began his schooling in the Grammar School in Dalkeith. However, when he was six years old his father died and Peter, with his two sisters and his mother, moved to Edinburgh to live with an uncle John Ronaldson. An Edinburgh banker, John Ronaldson was nevertheless interested in science, in particular in astronomy, geology and with the newly invented photography. He soon interested his young nephew Peter in these subjects and it is fair to say that Peter's interest in science was a direct consequence of his uncle's enthusiasm for the sciences.

When the family moved to Edinburgh Peter, of course, had to leave his school in Dalkeith. He next attended a private school in Circus Place Edinburgh, then in 1841, when he was ten years old, he entered Edinburgh Academy. Lewis Campbell, who later became the professor of Greek at the University of St Andrews, and James Clerk Maxwell were one year above Tait at the Academy. In fact Maxwell was slightly younger than Tait so the difference of one year certainly did not reflect their respective ages.

Tait was top of his class in each one of his six years at Edinburgh Academy. His early interests, however, were not in science but rather in classics. By his fourth year at the Academy mathematics had become his real love and that was the subject in which he really excelled. In 1846 he was placed first in the mathematics section of the Edinburgh Academical Club Prize which was no mean achievement given that he beat Lewis Campbell, who was placed second, and Maxwell who was placed third. In 1847, Tait's final year at Edinburgh Academy, Maxwell had his revenge since he was placed first for the Edinburgh Academical Club Prize with Tait second.

At the age of 16, in November 1847, Tait entered the University of Edinburgh. Maxwell entered Edinburgh University at the same time at Tait and together they attended the second mathematics class taught by Kelland and the natural philosophy (physics) class taught by James David Forbes. Tait remained at Edinburgh University for only one year before entering Peterhouse, Cambridge in 1848. There he was tutored by William Hopkins through what was a remarkable undergraduate career. In January 1852, at the age of twenty, he graduated as senior Wrangler in the Mathematical Tripos. This means that he was placed first among the First Class degrees in mathematics awarded by Cambridge in that year. He was also the first Smith's prizeman.

Maxwell followed Tait to Peterhouse in 1850 but transferred to Trinity where he believed that it was easier to obtain a fellowship. Another fellow student and friend of Tait's was William Steele who was in the same year as Tait and graduated as Second Wrangler. Tait won a Fellowship at Peterhouse and, in addition to coaching undergraduates for the Tripos, he began to collaborate with Steele in writing a text Dynamics of a particle. Tragically Steele died before much progress had been made with writing the book but Tait continued with the project and generously published the book under their joint authorship despite having written most of it himself. It was published in 1856.

In September 1854 Tait took up an appointment as professor of mathematics at Queen's College, Belfast. A number of the colleagues and friends he made in Belfast were to have a very significant effect on Tait's career. One of these was Thomas Andrews and the two collaborated in experiments to determine the density of ozone and also the affects of passing electrical discharge through oxygen and other gases. Tait had not been involved in experimental work up to this time and it is certainly due to the influence of Andrews that he added this interest to his growing range of skills. This research carried out with Andrews took Tait towards chemistry and this was a subject he retained an interest in through his career.

Another friendship of real significance was that with Hamilton. Tait had read Hamilton's Lectures on quaternions in 1853 while he was still at Cambridge but although the topic fascinated him he was more taken up with physical applications of mathematics at the time and did not pursue the topic at that stage. Then in July 1858 Tait read a paper by Helmholtz in Crelle's Journal on the motion of a perfect fluid. Helmholtz's paper Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen Ⓣ began by decomposing the motion of a perfect fluid into translation, rotation and deformation. Tait saw that using quaternions he could express the fluid velocity as a "vector function". It was the physical insight which Hamilton's quaternion differential calculus then gave which impressed Tait and he began to work hard developing a physical theory.

Tait began to correspond with Hamilton in August 1858 and, in reply to Hamilton's question as to how he had started to work with quaternions, Tait wrote to Hamilton on 7 December 1858 (see for example [7]):-

... it was only in August last that I suddenly bethought me of certain formulas I had admired years ago on page 610 of your Lectures - and I thought (and still think) likely to serve my purpose exactly. (The matter which more immediately suggested this to me was a paper by Helmholtz in Crelle's Journal (Vol. LX) which I was reading in July last as soon as we received it ... The title (in German) I forget - but a manuscript translation of my own which I now have beside me is headed "Vortex motion" ... ).

If Tait's friendship with Hamilton was to prove important for his future research, then other friendships which Tait formed were important in his family life. Two of his friends at Peterhouse were sons of the Rev James Porter and through them Tait met their sister, Margaret Archer Porter, whom he married in Belfast on 13 October 1857.

The Chair of Natural Philosophy at the University of Edinburgh became vacant in 1859, J D Forbes having moved to the University of St Andrews. Tait was a candidate for the chair but so was Maxwell who had been forced to seek another post when Marischal College and King's College in Aberdeen combined. Routh, who had been First Wrangler at Cambridge in Maxwell's year, was also a candidate but the real competition was always going to be between Tait and Maxwell. Tait won despite Maxwell's outstanding scientific achievements. When the Edinburgh paper, the Courant, reported the result it noted that Tait had been chosen in preference to Maxwell since:-

... there is another quality which is desirable in a Professor in a University like ours and that is the power of oral exposition proceeding on the supposition of imperfect knowledge or even total ignorance on the part of pupils.

The claim that Tait was the better person to teach poorly qualified pupils was certainly a fair one and, of course, Tait's personality meant that he made a stronger impression on the appointing committee rather than the much more reserved Maxwell.

By the time he arrived in Edinburgh in 1860 Tait was making strong contributions in applying Hamilton's quaternions. In the year he took up the chair of Natural Philosophy at Edinburgh he published Quaternion investigations connected with electro-dynamics and magnetism in which he reworked Helmholtz's hydrodynamic- electromagnetic analogy in the language of quaternions. As Epple writes in [11]:-

... not only quaternion analysis profited from acquiring a new. physical meaning. Quaternionic formulas also helped to grasp physical situations which could be described in terms of fluid motion more easily.

By 1863 when he published Note on a quaternion transformation in the Proceedings of the Royal Society of Edinburgh, Tait claimed that:-

... the next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions.

Hamilton died in 1865 and Tait took over the crusade to give quaternions a leading role in mathematical physics. Among the many contributions he made to the topic we should mention his two important texts Elementary Treatise on Quaternions (1867), and Introduction to Quaternions (1873). Maxwell was impressed by Tait's many works on physical applications of quaternions and wrote in a letter to William Thomson in 1871:-

You should let the world know that the true source of mathematical methods as applicable to physics is to be found in the Proceedings of the Royal Society of Edinburgh. The volume- surface- and line- integrals of vectors and quaternions and their properties as in the course of being worked out by Tait is worth all that is going on in other seats of learning.

Despite his intense work on quaternions, Tait was involved in many other activities. In 1862 he had published joint work with James A Wanklyn on electricity produced during evaporation and during effervescence. Three years later he published a paper on the motion of iron filings on a vibrating plate which was subjected to a magnetic field. In 1866 he started a joint project with the physicist Balfour Stewart on heating a disk which was rapidly rotating in a vacuum. This was a topic Tait came back to on several occasions throughout his career. Then in 1867 he published, in addition to the treatise on quaternions, his translation of Helmholtz's 1858 article and also the Treatise on Natural Philosophy for which he may be best known.

In 1861 Tait had been working on a text on mathematical physics. His friend William Thomson (later Lord Kelvin):-

... to my great delight offered to join.

The two intended to write a two volume work and Treatise on Natural Philosophy (1867) was to be the first of the volumes. However the second volume was never written. The treatise, known as T & T', was written mainly by Tait who seemed to find time despite his numerous other activities, while Thomson found that his many other activities prevented him finding as much time as Tait to work on the book. Hamilton Dickson writes in [10]:-

The work was epoch-making, and created a revolution in scientific development. For the first time T & T', as the authors called themselves, traced to Newton the concept of the 'conservation of energy' which was just then obtaining recognition among physicists, and they showed once and for all that 'energy' was the fundamental physical entity and that its 'conservation' was its predominant and all-controlling property.

We have already detailed major achievements for Tait dated 1867 but there is one further event of that year which we should mention. Helmholtz, in his 1858 paper, described the theoretical behaviour of vortex rings. He claimed that two interacting rings would change size and velocity as they interacted but would retain their ring shape. Tait verified Helmholtz's theoretical claims with experiments with smoke rings in 1867. He used two boxes each with a rubber diaphragm which shot out white smoke rings when the diaphragm was struck. Thomson wrote to Helmholtz on 22 January 1867:-

... a few days ago Tait showed me in Edinburgh a magnificent way of producing [vortex rings]. We sometimes can make one ring shoot through another, illustrating perfectly your description; when one ring passes near another, each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settles again into its circular form. ... The vibrations make a beautiful subject for mathematical work.

These experiments were to have a major influence on Thomson who saw the permanence of form as a possible explanation for atoms and therefore explain the way that the different elements could be built. Tait was not convinced by Thomson's idea at first, rightly so of course since, although a beautiful idea, it is quite wrong. The idea led Tait, Thomson and Maxwell to begin to work on knot theory since the basic building blocks, in Thomson's vortex atom theory, would be the rings knotted in three dimensions. By Helmholtz's theory of a perfect fluid, these knotted rings, although they could be distorted, would retain the 'same knot' as a circular knotted piece of string that can be moved around yet the form of the knot remains an invariant.

Tait, Thomson and Maxwell exchanged letters in which they invented many topological ideas as they looked at knots. Soon they discovered Listing's 1847 contributions to knot theory. Tait, although at first unconvinced by Thomson's vortex atom theory, began to include the theory in his lecture courses at Edinburgh in the early 1870s and he gave popular lectures describing the theory. In 1876 Tait began an intense study of knots, attempting to classify them. He published seven papers on knots in the Proceedings of the Royal Society of Edinburgh in the academic year 1876-77.

Tait considered alternating knots, namely those which when traversing the projection in 2-dimensional space the crossings go alternately over and under. He labelled the $n$ crossings of such a knot $A, B, C, ...$ and then the knot would be described by the sequence of crossings of length 2$n$ where each of $A, B, C, ...$ occurred exactly twice when the knot was traversed. There were then two basic problems to solve. Firstly which sequences of the above type corresponded to a knot, and secondly how could it be determined when two knots described by such sequences were the same.

Without any rigorous theory, which would have been well beyond nineteenth century mathematics, Tait began to classify knots using his mathematical and geometrical intuition. By 1877 he had classified all knots with seven crossings but he stopped there. One of the problems he considered after that was the colouring of graphs since he claimed to have a correct proof of the four colour theorem. His proof is fallacious and, sadly, he did not relate colouring of graphs to the knot theory he had considered a few years earlier. Another topic which he had worked on over a number of years was the results of the Challenger expedition on deep sea temperatures. In 1881 Tait published an important paper on the topic in which he showed how to correct the temperature readings because of the high pressures on the thermometers.

He returned to the topic of knots in his address to the Edinburgh Mathematical Society in 1883:-

We find that it becomes a mere question of skilled labour to draw all the possible knots having any assigned number of crossings. The requisite labour increases with extreme rapidity as the number of crossings is increased. ... I have not been able to find time to carry out this process further than the knots with seven crossings. ... It is greatly desired that someone, with the requisite leisure, should try to extend this list, if possible up to 11 ...

Kirkman read the text of Tait's address and began to work on classifying knots with more than seven crossings. He sent Tait his results on knot projections with up to nine crossings in May 1884 but he had not looked at the problem of deciding which of the projections led to equivalent knots. Tait worked on this side of the problem and, considering only alternating knots, solved the equivalence problems within a few weeks. Tait seemed to know how to tell whether two knots were equivalent without rigorous methods. He states this quite clearly in the paper he wrote tabulating the knots where he says that his methods have:-

... the disadvantage of being to a greater or less extent tentative. Not that the rules laid down ... leave any room for mere guessing, but they are too complex to be always completely kept in view. Thus we cannot be absolutely certain that by means of such processes we have obtained all the essentially different forms which the definition we employ comprehends.

Despite the problems Tait knew exactly what he was doing for, remarkably, his tables are correct. When Kirkman sent him all knot projections with 10 crossings in January 1885 again Tait found all in equivalent knots. The tables were printed in September 1885 and again they are completely correct. By then he had received from Kirkman 1581 knot projections with 11 crossings and this time Tait felt that he did not have the time to solve the equivalence problem for these.

It would be quite impossible in an article of this length to cover all the topics which Tait worked on. Knott [7] lists 365 papers and 22 books written by Tait. We will mention two final topics which he worked on after ending his work on knots. Thomson suggested that he work on the kinetic theory of gases and between 1886 and 1892 Tait published more than 20 papers on the topic. In this work he gave what Thomson considered the first proof of the Waterston-Maxwell equipartition theorem.

Tait also wrote a classic paper on the trajectory of golf balls (1896). The subject of golf was one of great interest to Tait. Of his four sons, the third was Frederick Guthrie Tait. He became the leading amateur golfer in 1893 and won the The Amateur Championship in 1896 and 1898. He finished third in The Open Championship in 1896 and 1897 and was leading amateur in the Open six times. Freddie Tait, as he was known in the golfing world, has a street in St Andrews named after him which is not far from my [EFR] home. Freddie was a military man in the Black Watch. He gave up his golf when he volunteered to serve in the Boer War in 1899. He was wounded at Magersfontein on 19 December 1899 and killed during fighting at Koodoosberg on 7 February 1900.

A deeply religious man, Tait wrote, with the physicist Balfour Stewart, The Unseen Universe (1875):-

... to overthrow materialism by a purely scientific argument.

Because of the public demand, he wrote a sequel Paradoxical Philosophy (1878).

We have painted a very positive picture of Tait in the details we have given above. This is right for he deserves no less. However there was another side to his character which we should mention. He became involved in many arguments with his fellow scientists and at least twice engaged in very public arguments. Tait was prone to let his heart rule his head in such situations and he often came of worse in the scientific debate. One of his disputes was with Heaviside and Gibbs whose vector methods he argued vigorously against over a long period. Certainly Tait came off worst in this arguments, perhaps his heart was too set on quaternionic methods to allow his head to see the importance of the ideas of Heaviside and Gibbs.

Another bitter dispute was with Clausius and Tyndall. Tait was patriotic to the extent that he would let such considerations prejudice his view of science. The dispute began over who was the first to propose the equivalence of work and heat. Tait and Tyndall began an argument over whether Joule or Julius Robert Mayer von Mayer had priority. Tait wrote a highly prejudiced account of the history of thermodynamics which was stupidly pro-British and Tyndall was right to be offended. Then Hopkins stumbled into the controversy when Tyndall had asked him to send him all von Mayer's papers but he was as pro-German as Tait was pro-British when he published an article in 1868 stating that not only did von Mayer have priority but so did the German nation. A more bitter dispute between Tait and Clausius began in 1872 when Maxwell published his Theory of Heat. Clausius stated that the British were trying to claim more than they deserved for the theory of heat which, given Tait's writing, was a fair comment. Maxwell, however, had over a number of years fully recognised Clausius's contribution, unlike Tait with his prejudiced approach.

Of course Tait's patriotism also meant that he was a devoted supporter of the Royal Society of Edinburgh which he served faithfully from the time he was elected a Fellow shortly after being appointed to the chair in Edinburgh. He served the Society as General Secretary for 22 years from 1879 until 1901. He won the Gunning Victoria Jubilee Prize and twice the Keith prize from that Society. Although never elected a Fellow of the Royal Society of London, he did have the distinction of receiving that Society's Royal Medal in 1886. Other honours given to Tait included the award of honorary degrees by the University of Glasgow and the University of Ireland, as well as being elected to honorary membership of the academies of Denmark, Holland, Sweden and Ireland.

At a ceremony in Peterhouse to present a portrait of Tait, Lord Kelvin spoke of his friend:-

I remember Tait once remarking that nothing but science is worth living for. It was sincerely said, but Tait himself proved it to be not true. Tait was a great reader. He would get Shakespeare, Dickens, and Thackeray off by heart. His memory was wonderful. What he once read sympathetically he ever after remembered. Thus he was always ready with delightful quotations, and these brightened our hours of work. For we did heavy mathematical work, stone breaking was not in it.

In [3] he is described as follows:-

His familiar figure was marked by a certain eccentricity, or carelessness of dress, and some of his intimate friends can scarcely remember ever to have seen him in a dress suit. Dining out was indeed an abomination in his eyes, unless it were informally in the company of one or two kindred spirits.