奥古斯塔斯·德摩根Augustus De Morgan

奥古斯塔斯·德摩根的父亲，弗瑞兹·约翰 德摩根（1771年10月5日 - 1816年11月27日），是马德拉斯土著步兵的中校。他出生并在印度服役于第22龙骑兵卫队，并于1798年在锡兰科伦坡与Elizabeth Dodson结婚。Elizabeth是伦敦海关的约翰 Dodson的女儿，也是James Dodson（1705-1757）的曾孙女，后者于1742年出版了The Anti-Logarithmic Canon. Being a table of numbers consisting of eleven places of figures, corresponding to all Logarithms under 100,000, with an Introduction containing a short account of Logarithms。约翰和Elizabeth 德摩根有七个孩子：约翰 德摩根（生于1799年5月16日，在1804年Prince of Wales从印度返回途中失事时去世）；James 德摩根（也在1804年Prince of Wales失事中去世）；Eliza 德摩根（生于1801年9月27日）；Georgina 德摩根（生于1805年3月，卒于1812年）；德摩根（生于1806年6月27日，本传记的主题）；George 德摩根（生于1808年7月15日，成为律师，卒于1890年）；和Campbell Grieg 德摩根（生于1811年11月22日，成为著名外科医生，卒于1876年4月12日）。

德摩根出生后不久，双眼都感染了印度“眼炎”，随后失去了右眼的视力。他的一只眼睛保住了，但一只眼睛失明了。他于1806年10月20日在印度马德拉斯的圣乔治堡受洗。七个月大时，他随父母以及姐妹Eliza和Georgina返回英国。一家人乘坐Duchess of Gordon号船航行到英国，这是船队中的众多船只之一，并在伍斯特定居。德摩根的父亲于1808年独自返回印度，但于1810年返回英国。他们先后住在阿普尔多尔、比迪福德和巴恩斯特珀尔，都在德文郡。1812年，一家人在萨默塞特郡的汤顿定居。约翰德摩根返回印度马德拉斯，但于1816年患上肝脏疾病，在返回英国的途中于圣赫勒拿岛去世。父亲去世时德摩根10岁，但在他晚年列出的教师名单中，他把父亲列为他的第一位老师。

德摩根的学校教育始于Barnstaple，在那里由Williams小姐教他阅读和写作；随后在Taunton，1813—1814年由Poole太太教他阅读、写作和算术，接下来几年由J Fenner牧师教他希腊文和拉丁文。后来在Blandford，由T 约翰·梅纳德·凯恩斯牧师教他；之后在Taunton，由H Barker牧师教他拉丁文、希腊文、欧几里得几何和代数。最后，他进入布里斯托尔附近Redland的Parsons先生学校，从十四岁读到十六岁半。在Parsons先生的学校里，德摩根并不出众，而且由于他的身体残疾[23]：-

……他不参加其他男孩的运动，甚至成为一些同学残酷恶作剧的受害者。

关于德摩根在Parsons先生学校期间的更多细节，见THIS LINK。

在详细介绍德摩根的教育时，我们尚未提及他的宗教教育。然而，这非常重要，因为他所受的严格训练使他对教会产生反感，尽管他仍是一名虔诚的基督徒。他的母亲希望他成为教会中的福音派牧师，并为此向他施压，要他上大学学习。他的校长帕森斯先生也向他施压，要他在大学学习古典学，但德摩根的热爱是数学。

德摩根于1823年2月进入剑桥大学三一学院，时年16岁，在那里由乔治·皮科克和威廉·休厄尔教他数学——三人成为终身朋友。他的学院导师是J P Higman，他还听了乔治·比德尔·艾里、Henry Coddington（1798—1845）和Henry Parr Hamilton（1794—1880）的课。尽管德摩根的本科生涯是成功的，然而他并没有以人们可能预期的方式大放异彩，这必定有若干原因。他的母亲在宗教方面给他施加压力，这给他带来了困难。他可能把太多时间花在古典学学习上，至少最初几年肯定如此，而且他的健康状况有时很差。他有通宵学习的习惯，然后起得很晚，这可能加剧了他的健康问题。他也不确定自己的学习应当通向何方，在最后两年里他认真考虑过从医。我们在上文注意到，他的弟弟Campbell Grieg 德摩根确实走上了从医之路。

德摩根在学生时代最大的消遣是吹长笛，他吹奏水平很高。他的许多朋友都喜欢听他吹长笛，并会请他演奏。

此处一段未译出，以下为英文原文He received his B.A. in 1827, being Fourth 数学荣誉学位考试一等及格者（Wrangler） in the Mathematical tripos. Henry Percy Gordon (1806-1876) was Senior Wrangler; he had a career in law. Thomas Turner (1804-1883) was Second Wrangler and First Smith's Prizeman. Turner also had a career in law but was an early fellow of the Royal Astronomical Society and had a lifelong interest in astronomy. Anthony Cleasby (1804-1879) was Third Wrangler; he also had a career in law. Although the three above De Morgan were undoubtedly extremely able, as their subsequent careers showed, nevertheless it seems certain that they lacked De Morgan's mathematical abilities. Certainly another factor here was De Morgan's dislike of the tripos type examination where cramming was the key to success rather than demonstrating originality [15]:-

这位年轻数学荣誉学位考试优胜者的名次，虽然未能彰显他真正的实力或他在数学研究方面的非凡天赋，却足以使他获得一份研究员职位；而且，若非他出于良心的顾虑，不愿签署当时要求文学硕士学位的获得者以及所有学院研究员都必须签署的测试条款，他无疑会在大学的高墙之内找到一个志同道合的工作领域。

由于文学硕士学位要求通过一项神学测试，而德摩根尽管是英格兰教会成员却对此强烈反对，他在剑桥无法继续深造，因为没有文学硕士学位就没有资格获得研究员职位。1826年，他回到伦敦的家中，尽管他担心自己的良心会使他成为一个糟糕的律师，他还是进入了林肯律师学院学习法律，准备当律师。他在一封信中[7]明确表达了自己真正的兴趣所在：-

你似乎以为我是出于自愿才去当律师的。事实是，在所有被称为有学问的职业中，律师行业对我最为开放；但我的选择是，只要科学能养活我，我就坚持搞科学。我很高兴自己能入睡而不至于梦见一份“五联契据”或诸如此类的东西挡在我和《天体力学》之间，同时心里清楚这梦一定会成真。

1827年（时年21岁），他申请了新成立的伦敦大学的数学讲席，尽管没有任何数学出版物，他还是被任命了。1828年2月23日，德摩根成为伦敦大学第一位数学教授；他发表了他的就职演讲On the study of mathematics。在这次演讲中[27]：-

……德摩根将数学描述为对关于清晰明确观念的自明定律或公理的演绎研究。……他赞扬了洛克的《人类理解论》，并声称：“众所周知，任何人最初获得的观念都源自周围物体的形状或数目。从物质世界的表象中，人们收集到某些明确的观念，这些观念虽然其原型在自然界中并不真实存在，却是我们心灵中所包含的最清晰、最确定的观念。”

德摩根写道[7]：-

这篇题为《论数学的爱德华·斯图迪》的讲演，对这门学问及其对心智的影响所持的视野，远比其标题本身所暗示的更为宽广。它是一篇关于知识之进步、知识之必要、人人有权获得所能给予他的尽可能多的知识，以及推理能力的培养在智力发展中应占何种地位的论文。它不仅是一篇关于心智教育的论述，也是一篇关于心智本身的论述。

德摩根说，教学是学习一门学科的最佳途径。他[15]：-

……开始以比他所受教的方式更好的目的自学，正如每一个不是傻瓜的人，在开始教别人时都会做的那样，不管他以前的老师是什么样的人。

1828年，德摩根出版了The Elements of Algebra，这是他翻译的Pierre Louis Marie Bourdon（1779-1854）所著Élémens d'algèbre前三章的英译本。这本书“是为伦敦大学的学生使用而设计的”。在书中，德摩根写道（日期为1828年8月）：-

以下译文是为伦敦大学中那些可能无法阅读法语，或不想将代数学习推进到二次方程之后的学生准备的。译者认为，原著特别适合初等教学，因为它精心地从基本原理推导每一条规则，并区分约定俗成的结果与论证得出的结果。本可尝试翻译全书，但考虑到目前每一个想要获得相当程度数学知识的人都必须熟悉法语；只有对这些人来说，整本书才是必要的。

德摩根非常热衷于区分定理与问题，并在第一页加上了以下“译者注”：-

The first is a theorem, the second a problem.

1. The greater of two numbers is equal to half their sum added to half their difference.
2. What two numbers are those whose sum is 20 and whose difference is 10.

更令人惊讶的是德摩根关于负数的注释，看来他并不真正相信：-

请注意，负量只是指要减去的量；而像
$-a - a = -2a$
这样的表达式，意思是：从任意数中连续两次减去a，与一次减去$2a$的结果相同。为了防止对负号含义产生错误理解，学生应当养成把诸如
5 - 8 = -3
这样的表达式翻译成普通语言的习惯，它的意思是：先加5再减8，依次进行，等价于减去3……

1829年夏天是在巴黎度过的，在那里他结识了Jean Hachette、让-巴蒂斯特·毕奥等人。在接下来的几年里，他与Hachette互通了几封信，直到1834年Hachette去世。1830年，德摩根出版了Elements of Arithmetic。他写道：-

这部小著作试图向年轻学生提供算术的常用法则，并附以推理，学生必须使自己的头脑习惯于这种推理，才能在任何科学中取得进步。

我或许可以从经验出发，谈谈大多数青年在学习几何和代数之前所获得的那种算术知识的性质。但既然几乎所有人都同意，这门科学不应像在这个国家那样，被贬低为一堆死记硬背的规则——其中一半除了在商业事务中有用外别无他用，甚至在那里也很少有用——我将就这本书应当如何学习发表一点看法。

为了避免代数语言的概括性——初学者的头脑无法把握它——必须把每个证明限制在一个特定情形中；也就是说，在一些特定的数上展示那些在代数中借助代表数的字母一次性断言为对所有数都成立的真理。从所选择的情形中，引出一条被假定始终成立的法则。这种推理并非严格合乎逻辑；但必须记住，学生有能力通过在每个证明中使用与正文不同的数，使自己确信所述内容的普遍真理性。这是我建议他做的：如果他省略了这一练习，他就没有公正地检验这个主题。

德摩根于1831年因原则问题辞去了他的讲席。一些德摩根的传记称他辞职是因为一位同事教授被解职。尽管这是真的，但原因要更为复杂一些，并涉及伦敦大学被治理的整个方式。教授们可能被一个几乎没有学术专长的管理机构无正当理由地解职，这是德摩根强烈感到不满的事情。他写道（见[7]）：-

遵照我与您会面时您所表达的愿望，我向您陈述我对一个与大学福祉最为密切相关的问题所持的看法，即教授们在大学中应当享有的地位。这个问题至关重要，因为教育秩序和未来教授中体现的功绩，以及公众对他们的评价，都取决于这个问题将如何解决。

为了吸引有品格的人担任大学教授，必须使这些职位具有高度的独立性和受人尊敬的地位。任何对自己有正确认识的人，只要他出现在学生面前的身份中有任何东西会引起除了最完全的尊重之外的其他感受，他就不会面对一班学生。学生们都知道大学中有一个高于教授的机构；他们也应当知道这个机构尊重教授，并且只要教授履行职责，大学的基本法律就会保护他，正如在行为不当或疏忽职守时这些法律必定会导致他被免职一样。除非学生们确信这一点，否则他们会把教授的地位看作是非常可疑的体面，而他们唯一的错误在于：在这种情况下根本不存在任何可疑之处。

这些问题从他第一次被任命时就存在，最终因⟦N3⟧（1791-1851）被解职而达到顶点，他是伦敦大学第一位解剖学教授。

关于进一步细节，包括德摩根的辞职信，见THIS LINK。

辞职后，德摩根从位于吉尔福德街的家中搬到上高尔街5号。此时有一个显而易见的问题，即他在五年没有工作的情况下是如何在经济上维持自己的？看来他是通过接收私人学生以及向各家公司提供精算建议来挣钱的。伦敦大学任命乔治·詹姆斯·佩利·怀特接替德摩根担任数学教授。怀特与德摩根相似，也曾是三一学院的人，拥有相同的导师和推荐人；事实上，他明显是最佳候选人。

也许德摩根在此期间承担的最重要工作是他为皇家天文学会所做的工作。他于1828年5月9日当选为会士，并于1831年至1839年担任该学会秘书（1847年至1855年再次担任）[5]：-

……这种成果发表不够迅速的问题，因所有宣读论文的出色且相当详细的摘要而变得不那么有害，这些摘要此时成为《月刊通讯》的固定特色。……毫无疑问，德摩根作为1831年至1839年的秘书，对学会出版物中这一非常有用的部分应享有相当大的功劳。终其一生，德摩根始终对学会保持热切的关注，并定期参加会议。……他坚决拒绝了主席一职，他认为这一职位不应由一位不是天文学活跃研究者的人担任。……他个人的才华，他既广博又精细、涵盖历史与当代的学识，他对当时最优秀数学的掌握——远超同时代人——使他的名声在 intervening 的数十年间有增无减。但在他与理事会的关系中，我们关注的是他个人的一面，那种对原则的执着热情，对他来说胜过任何回报或成功。

1836年，乔治·怀特在一次划船事故中去世后，他再次被任命为讲席教授，并一直担任到1866年，那时他第二次辞职，同样是因为原则问题。

关于他1836年任命的详情，见THIS LINK。

关于他1866年辞职的详情，见THIS LINK。

德摩根于1837年8月3日与Sophia Elizabeth Frend（1809-1892）结婚。德摩根十年前通过与其父亲William Frend的友谊结识了Sophia，后者在Principles of Algebra 工作。Frend曾出版Principles of Algebra （1796年），附有Francis Maseres的附录；Frend拒绝使用负量。由于他的强烈观点，德摩根不想要带有通常婚礼仪式的教堂婚礼，因此他们在登记处由Rev Thomas Madge主持结婚。仪式形式省略了婚礼服务中‘丈夫和妻子的职责’部分。德摩根和Sophia 德摩根有七个孩子：Elizabeth Alice 德摩根（1838年6月出生）；William Frend 德摩根（1839年11月出生）；George Campbell 德摩根（1841年10月出生）；Edward I 德摩根（1843年6月出生）；Anne Isabella 德摩根（1845年2月11日出生）；Helena Christiana 德摩根（1848年3月20日出生）；Mary Augusta 德摩根（1850年2月24日出生）。

1838年，他定义并引入了术语‘数学归纳法’，将此前使用不清晰的过程置于严格的基础上。该术语首次出现在德摩根的文章Induction (Mathematics)中，发表于Penny Cyclopedia。（多年来，他为Penny Cyclopedia撰写了712篇文章。）Penny Cyclopedia由有用知识传播协会出版，该协会由创立伦敦大学的同一批改革者建立，该协会还出版了德摩根的著名著作The Differential and Integral Calculus （1836年）。在此书中，他：-

... 努力使极限成为科学的唯一基础，完全不借助级数理论或代数表达式。

1849年，他出版了Trigonometry and double algebra，其中给出了复数的几何解释。他在序言中写道：-

读者面前的这部著作完全是新的，绝不是我在1837年就同一主题出版的著作的第二版。它由两本书组成。在第一本中，我努力为具备足够算术和代数知识的学生……提供三角学作为代数的一个分支和高等数学基础组成部分的视角。在第二本中，我以纯粹符号特征给出了基本视角，并应用了那种几何基础的意义，从而解释每一个符号。

他认识到代数的纯粹符号性质，并且意识到存在普通代数之外的代数。他引入了德摩根的定律，而他最大的贡献是作为数学逻辑的改革者。

德摩根与查尔斯·巴贝奇通信，并为阿达·洛夫莱斯提供私人辅导，据称后者为查尔斯·巴贝奇编写了第一个计算机程序。他还与威廉·哈密顿通信，并像威廉·哈密顿一样试图将双重代数扩展到三维。在给威廉·哈密顿的一封信中，德摩根写到了他与威廉·哈密顿和威廉·哈密顿从男爵的通信。他写道：-

须知，我发现你与另一位W H先生相对于我而言是互反极点（在理智和道德上，因为那位苏格兰从男爵是一头北极熊，而你，我本要说，是一位极地绅士）。当我向爱丁堡寄去一点研究时，那里的W H说我抄袭了他。当我寄给你一份时，你从我这里拿走，一眼就将其推广，如此推广后赠予整个社会，并使我成为已知定理的第二发现者。

1864年，他是伦敦数学会的联合创始人之一，提议了其名称，并成为其第一任主席。由于其与本档案馆的相关性，我们引用他1865年1月16日在‘学会第一次会议’上所作主席致辞的一部分：-

我说，任何艺术或科学，除非与过去时代人的心智相联系来研究，否则就不是自由艺术或自由科学。令人惊讶的是，数学家们谈论数学的方式多么奇怪，因为他们不了解自己学科的历史。通过断言他们认为是事实的东西，他们以这种方式扭曲了其历史。在每个人的观念中，都有某种特定的命题序列，他心中有其自己的顺序，并想象这一序列存在于历史中；他自己的顺序就是命题相继演化出来的历史顺序。数学家需要知道数学不同分支中发明的过程是什么；他想看到艾萨克·牛顿通过约翰·沃利斯已经给出的更高定理的提示，引出并演化出二项式定理。如果他要以最有可能引导他成功的方式指导自己的研究，他必须看到较低命题不断从较高命题演化出来的奇特方式。

德摩根的儿子乔治，一位非常能干的数学家，成为伦敦数学会的第一任秘书。德摩根从未成为伦敦皇家学会的会士，因为他拒绝让人提名他。他还拒绝了爱丁堡大学的荣誉学位。他被托马斯·阿彻·赫斯特如此描述：-

我担心德摩根先生是一个枯燥教条的学究，尽管他的能力毋庸置疑。

Macfarlane 指出 [23]：-

……德摩根 自认为是“无归属的英国人”，既非英格兰人、苏格兰人、威尔士人，也非爱尔兰人。

他还写道 [23]：-

他不喜欢这个国家，当他的家人在海边享受时，科学界人士在乡间参加 英国协会 的会议度过愉快时光，他却留在首都炎热多尘的图书馆里。……他与物理哲学家没有共同的想法或共鸣。他的态度无疑是由于他身体虚弱，这使他既不能成为观察者，也不能成为实验者。他从未在选举中投票，也从未参观过下议院、伦敦塔或威斯敏斯特教堂。

德摩根 一直对奇特的数字事实感兴趣，并在 1864 年写道，他拥有在 $x^{2}$ 年时为 $x$ 岁的殊荣（他在 1849 年时 43 岁）。任何出生于 1980 年的人都可以在 2025 年声称同样的殊荣。

关于德摩根最后几年的详情，见THIS LINK。

他去世五天后，即1871年3月23日，举行了葬礼，他被安葬在伦敦肯辛顿和切尔西的肯萨尔绿地的诸灵教堂。

Augustus De Morgan's father, John De Morgan (5 October 1771 - 27 November 1816), was a Lieutenant-Colonel in the Madras Native Infantry. He was born and served in India in the 22 Dragoon Guards and married Elizabeth Dodson in 1798 at Colombo, Ceylon. Elizabeth was the daughter of John Dodson of the Custom House, London and the great-granddaughter of James Dodson (1705-1757) who published The Anti-Logarithmic Canon. Being a table of numbers consisting of eleven places of figures, corresponding to all Logarithms under 100,000, with an Introduction containing a short account of Logarithms in 1742. John and Elizabeth De Morgan had seven children: John Augustus De Morgan (born 16 May 1799 and died when the Prince of Wales was wrecked on its passage back from India in 1804); James De Morgan (also died in the 1804 wreck of the Prince of Wales); Eliza De Morgan (born 27 September 1801); Georgina De Morgan (born March 1805 and died in 1812); Augustus De Morgan (born 27 June 1806, the subject of this biography); George De Morgan (born 15 July 1808, who became a barrister and died in 1890); and Campbell Grieg De Morgan (born 22 November 1811, who became a famous surgeon and died 12 April 1876).

Augustus lost the sight of his right eye shortly after birth when both eyes were affected with Indian "sore eye". One of his eyes was saved but he became blind in one eye. He was baptised on 20 October 1806 at Fort St George, Madras, India. When seven months old, he returned to England with his parents, and his sisters Eliza and Georgina. The family sailed to England in the Duchess of Gordon, one of many ships in a convoy, and settled in Worcester. Augustus's father returned to India on his own in 1808, but returned to England in 1810. They lived at Appledore, then at Bideford, then at Barnstaple, all in Devon. In 1812 the family settled in Taunton in Somerset. John De Morgan returned to Madras in India but in 1816 became ill with a liver problem and died in St Helena on a return voyage to England. Augustus was 10 years old when his father died but, in a list of teachers made by him in later life, he gave his father as his first teacher.

De Morgan's schooling began in Barnstaple where he was taught reading and writing by Miss Williams, then in Taunton where, 1813-14, Mrs Poole taught him reading, writing and arithmetic and in the next couple of years the Rev J Fenner taught him Greek and Latin. Later in Blandford he was taught by the Rev T Keynes, then at Taunton, he was taught Latin, Greek, Euclidean geometry and algebra by the Rev H Barker. Finally he attended Mr Parsons' school, at Redland, near Bristol, where he studied from age fourteen to sixteen and a half. At Mr Parsons' school, De Morgan did not excel and, because of his physical disability [23]:-

... he did not join in the sports of other boys, and he was even made the victim of cruel practical jokes by some schoolfellows.

For further details of De Morgan's time at Mr Parsons' school, see THIS LINK.

What we have not mentioned when giving details of De Morgan's education is his religious education. This, however, was highly significant since the strict training he received put him off the Church, although he remained a committed Christian. His mother wanted him to become an Evangelical Minister in the Church and put pressure on him to study at university with this aim. His schoolmaster, Mr Parsons, put pressure on him to study classics at university, but De Morgan's love was mathematics.

De Morgan entered Trinity College Cambridge in February 1823 at the age of 16 where he was taught mathematics by George Peacock and William Whewell - the three became lifelong friends. His College tutor was J P Higman, and he also attended lectures by George B Airy, Henry Coddington (1798-1845), and Henry Parr Hamilton (1794-1880). Although De Morgan's undergraduate career was successful, nevertheless, he did not shine in the way one might expect and there must have been a number of reasons for this. His mother put pressure on him about religion which gave him difficulties. He probably devoted too much time to his study of Classics, certainly in his first years, and his health was poor at times. He had the habit of studying all through the night, then getting up very late which may have contributed to his health problems. He was also unsure of where his studies should lead and in his final couple of years he thought seriously about a medical career. We noted above that his younger brother Campbell Grieg De Morgan did follow a medical career.

Perhaps De Morgan's greatest relaxation while a student was in playing the flute which he did to a high standard. Many of his friends would love to listen to his flute playing and would ask him to play.

He received his B.A. in 1827, being Fourth Wrangler in the Mathematical tripos. Henry Percy Gordon (1806-1876) was Senior Wrangler; he had a career in law. Thomas Turner (1804-1883) was Second Wrangler and First Smith's Prizeman. Turner also had a career in law but was an early fellow of the Royal Astronomical Society and had a lifelong interest in astronomy. Anthony Cleasby (1804-1879) was Third Wrangler; he also had a career in law. Although the three above De Morgan were undoubtedly extremely able, as their subsequent careers showed, nevertheless it seems certain that they lacked De Morgan's mathematical abilities. Certainly another factor here was De Morgan's dislike of the tripos type examination where cramming was the key to success rather than demonstrating originality [15]:-

The place of the youthful wrangler, though it failed to declare his real power or the exceptional aptitude of his mind for mathematical study, would, however, have been sufficient to have secured for him a fellowship, and he, no doubt, would have found a congenial field of labour within the walls of his university, if his conscientious scruples had not prevented his signing the tests which at that time were required from those who took up their degree of M.A. as well as from all Fellows of Colleges.

Because a theological test was required for the M.A., something to which De Morgan strongly objected despite being a member of the Church of England, he could go no further at Cambridge being not eligible for a Fellowship without his M.A. In 1826 he returned to his home in London and, despite having doubts that his conscience would make him a poor lawyer, he entered Lincoln's Inn to study for the Bar. He made it clear where his real interests were in one of his letters [7]:-

You seem to fancy that I was going to the Bar from choice. The fact is, that of all the professions which are called learned, the Bar was the most open to me; but my choice will be to keep to the sciences as long as they will feed me. I am very glad that I can sleep without the chance of dreaming that I see an "Indenture of Five Parts," or some such matter, held up between me and the 'Mecanique Celeste', knowing all the time that the dream must come true.

In 1827 (at the age of 21) he applied for the chair of mathematics in the newly founded London University and, despite having no mathematical publications, he was appointed. On 23 February 1828, De Morgan became the first professor of mathematics at the London University; he gave his inaugural lecture On the study of mathematics. In this lecture [27]:-

... De Morgan described mathematics as the deductive study of self-evident laws or axioms concerning clear and distinct ideas. ... he praised Locke's 'Essay Concerning Human Understanding' and claimed: "It is notorious that the first ideas which any human being receives are derived either from the figure or number of the objects which surround him. From the appearances of the material world, certain distinct notions are gathered, which though their prototypes have no real existence in nature, are the clearest and most definite which our minds contain."

Sophia De Morgan writes [7]:-

This lecture 'On the Study of Mathematics' takes a much wider view of that study, and its effects upon the mind, than its title alone would imply. It is an essay upon the progress of knowledge, the need of knowledge, the right of everyone to as much knowledge as can be given to him, and the place in mental development which the culture of the reasoning power ought to hold. It is not only a discourse upon mental education, but upon mind itself.

Teaching was, De Morgan said, the best way to learn a subject. He [15]:-

... began to teach himself to better purpose than he had been taught, as does every man who is not a fool, when he begins to teach others, let his former teachers be what they may.

In 1828 De Morgan published The Elements of Algebra, his English translation of the first three chapters of Élémens d'algèbre by Pierre Louis Marie Bourdon (1779-1854). This book was "designed for the use of students in the University of London." In it, De Morgan writes (dated August 1828):-

The following translation has been prepared for the use of such students in the University of London as may not be able to read French, or do not desire to pursue their algebraical studies further than Equations of the Second Degree. The original work, in the opinion of the translator, is particularly well adapted for elementary instruction, on account of the care which is taken to deduce every rule from first principles, and to distinguish between the results of convention and those of demonstration. A translation of the whole would have been attempted, but or the consideration that at present every one who is desirous of attaining a considerable degree of mathematical knowledge must become acquainted with the French language; and it is to such only that the whole book would be necessary.

De Morgan is very keen to distinguish between a theorem and a problem and on the first page he added the following "translator's note":-

The first is a theorem, the second a problem.

1. The greater of two numbers is equal to half their sum added to half their difference.
2. What two numbers are those whose sum is 20 and whose difference is 10.

Much more surprising is De Morgan's note on negative numbers in which, it appears, he does not really believe:-

Observe, that by a negative quantity is only meant a quantity to be subtracted; and by such an expression as
          $-a - a = -2a$,
is meant that subtraction of a from any number twice following gives the same result as the subtraction of $2a$ once. To guard against erroneous ideas concerning the meaning of the negative sign, the student should accustom himself to translate into common language such expressions as
          5 - 8 = -3
which means, that the addition of 5 and the subtraction of 8, performed one after the other, is equivalent to the subtraction of 3 ...

The summer of 1829 was spent in Paris where he met Jean Hachette, Jean-Baptiste Biot among others. He exchanged several letters with Hachette over the next few years until Hachette's death in 1834. In 1830 De Morgan published Elements of Arithmetic. He wrote:-

This little work is an attempt to give the young student the common rules of Arithmetic, accompanied by the reasoning to which he must habituate his mind before he can make progress in any science.

I might speak from experience, of the nature of the arithmetical knowledge which most youths acquire before they commence the study of geometry and Algebra. But as almost all agree in opinion, that this science ought not to be, as it is in this country, degraded into a mass of rules learned by rote, one half of which are of no use but in commercial business, and rarely even there, I will proceed to make a remark on the manner in which this book should be studied.

In order to avoid the generalities of algebraic language, which the mind of a beginner cannot grasp, it is necessary to confine each demonstration to one particular case; that is, to show, on some particular numbers, those truths which, in Algebra, are asserted of all at once, by means of letters to stand for numbers. From the case which is chosen, a rule is drawn which is assumed to hold good always. This reasoning is not strictly logical; but it must be recollected, that the student has it in his power to convince himself of the universal truth of what is stated, by employing different numbers from those used in the text, in every demonstration. This is what I recommend him to do: if he omits this exercise, he does not give the subject a fair trail.

De Morgan was to resign his chair, on a matter of principle, is 1831. Some biographies of De Morgan state that he resigned because a fellow professor was dismissed. Although this is true, the reasons are somewhat more complex and involve the whole way in which the London University was governed. That the professors could be dismissed without good cause by a governing body which had little academic expertise was something that De Morgan felt strongly about. He wrote (see [7]):-

In compliance with the wish expressed by you when I had the honour of an interview with you, I lay before you the views which I entertain on a subject most essentially connected with the welfare of the University, viz., the situation which the Professors ought to hold in the establishment. This question is of the highest importance, inasmuch as upon the manner in which it shall be settled depends the order of education and merit which will be found among the Professors in future, and the estimation in which they will be held by the public.

In order to induce men of character to fill the chairs of the University, these latter must be rendered highly independent and respectable. No man who feels (rightly) for himself will face a class of pupils as long as there is anything in the character in which he appears before them to excite any feelings but those of the most entire respect. The pupils all know that there is a body in the University superior to the Professors; they should also know that this body respects the Professors, and that the fundamental laws of the institution will protect the Professor as long as he discharges his duty, as certainly as they will lead to his ejectment in case of misconduct or negligence. Unless the pupils are well assured of this they will look upon the situation of Professor as of very ambiguous respectability, and they will only be wrong inasmuch as there will be no ambiguity at all in the case.

These problems, which were there from his first appointment, came to a head with the dismissal of Granville Sharp Pattison (1791-1851), the first professor of anatomy at the London University.

For further details, including De Morgan's resignation letter, see THIS LINK.

After resigning, De Morgan moved from the family home in Guilford Street to 5 Upper Gower Street. There is an obvious question at this point, namely how did he support himself financially for five years without a job? It appears that he earned money by taking private pupils and by giving actuarial advice to various companies. The London University appointed George James Pelly White to succeed De Morgan as Professor of Mathematics. White was similar to De Morgan in having been a Trinity man with the same tutors and referees; in fact he stood out as clearly the best candidate.

Perhaps the most important work that De Morgan undertook during this period was his work for the Royal Astronomical Society. He had been elected a fellow on 9 May 1828 and served as secretary to the Society from 1831 to 1839 (again from 1847 to 1855) [5]:-

... this want of rapid publication of results was rendered less harmful by the excellent and fairly detailed summaries of all papers read, which now became a regular feature of the 'Monthly Notices'. ... there can be no doubt that De Morgan, who was Secretary from 1831-39, deserves a considerable share of the credit of this very useful part of the Society's publications. Throughout his life De Morgan continued to be warmly interested in the Society and was a regular attendant at the meetings. ... he firmly declined the office of President, which he did not think ought to be held by a man who was not an active worker in astronomy. ... His personal brilliance, his learning, at once extensive and minute, historical and modern, his hold on the best mathematics of the day, much in advance of his contemporaries, have made his name rather increase than diminish with the intervening decades. But in his relations to the Council it is his personal side that concerns us, that master passion for principle which was more than any reward or success for him.

He was appointed to the chair again in 1836, after George White died in a boating accident, and held it until 1866 when he was to resign for a second time, again on a matter of principle.

For details of his 1836 appointment, see THIS LINK.

For details of his 1866 resignation, see THIS LINK.

De Morgan married Sophia Elizabeth Frend (1809-1892) on 3 August 1837. De Morgan had met Sophia ten years earlier through his friendship with her father William Frend who worked at the Nautical Almanac. Frend had published Principles of Algebra (1796) with an appendix by Francis Maseres; Frend rejected the use of negative quantities. Because of his strong views, De Morgan did not want a Church wedding with the usual marriage ceremony so they were married in a Registry Office by the Rev Thomas Madge. The form of service omitted the 'duties of husbands and wives' part of the wedding service. Augustus and Sophia De Morgan had seven children: Elizabeth Alice De Morgan (born June 1838); William Frend De Morgan (born November 1839); George Campbell De Morgan (born October 1841); Edward I De Morgan (born June 1843); Anne Isabella De Morgan (born 11 February 1845); Helena Christiana De Morgan (born 20 March 1848); Mary Augusta De Morgan (born 24 February 1850).

In 1838 he defined and introduced the term 'mathematical induction' putting the process that had been used without clarity on a rigorous basis. The term first appears in De Morgan's article Induction (Mathematics) in the Penny Cyclopedia. (Over the years he was to write 712 articles for the Penny Cyclopedia.) The Penny Cyclopedia was published by the Society for the Diffusion of Useful Knowledge, set up by the same reformers who founded the London University, and that Society also published a famous work by De Morgan The Differential and Integral Calculus (1836). In this he:-

... endeavoured to make limits the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expressions.

In 1849 he published Trigonometry and double algebra in which he gave a geometric interpretation of complex numbers. He writes in the Preface:-

The work before the reader is entirely new, not being in any sense a second edition of that which I published on the same subject in 1837. It consists of two books. In the first, I have endeavoured to give the student who has a competent knowledge of arithmetic and algebra ... a view of trigonometry, as a branch of algebra and a constituent part of the foundation of higher mathematics. In the second, I have given an elementary view in its purely symbolic character, with the application of that geometrical basis of significance which affords explanation of ever symbol.

He recognised the purely symbolic nature of algebra and he was aware of the existence of algebras other than ordinary algebra. He introduced De Morgan's laws and his greatest contribution is as a reformer of mathematical logic.

De Morgan corresponded with Charles Babbage and gave private tuition to Ada Lovelace who, it is claimed, wrote the first computer program for Babbage. He also corresponded with Hamilton and, like Hamilton attempted to extend double algebra to three dimensions. In a letter to Hamilton, De Morgan writes of his correspondence with Hamilton and William Hamilton. He writes:-

Be it known unto you that I have discovered that you and the other Sir W H are reciprocal polars with respect to me (intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman). When I send a bit of investigation to Edinburgh, the W H of that ilk says I took it from him. When I send you one, you take it from me, generalise it at a glance, bestow it thus generalised upon society at large, and make me the second discoverer of a known theorem.

In 1864 he was a co-founder of the London Mathematical Society, suggesting its name, and became its first president. We quote, because of its relevance to this Archive, part of his President's Address of 16 January 1865 given at the 'First Meeting of the Society':-

I say that no art or science is a liberal art or a liberal science unless it is studied in connection with the mind of man in past times. It is astonishing how strangely mathematicians talk of the Mathematics, because they do not know the history of their subject. By asserting what they conceive to be facts they distort its history in this manner. There is in the idea of everyone some particular sequence of propositions, which he has in his own mind, and he imagines that the sequence exists in history; that his own order is the historical order in which the propositions have successively been evolved. The mathematician needs to know what the course of invention has been in the different branches of Mathematics; he wants to see Newton bringing out and evolving the Binomial Theorem by suggestion of the higher theorem which Wallis had already given. If he be to have his own researches guided in the way which will best lead him to success, he must have seen the curious ways in which the lower proposition has constantly been evolved from the higher.

De Morgan's son George, a very able mathematician, became the first secretary of the London Mathematical Society. De Morgan was never a Fellow of the Royal Society of London as he refused to let his name be put forward. He also refused an honorary degree from the University of Edinburgh. He was described by Thomas Hirst thus:-

A dry dogmatic pedant I fear is Mr De Morgan, notwithstanding his unquestioned ability.

Macfarlane remarks that [23]:- 

... De Morgan considered himself a 'Briton unattached' neither English, Scottish, Welsh or Irish.

He also writes [23]:-

He disliked the country and while his family enjoyed the seaside, and men of science were having a good time at a meeting of the British Association in the country he remained in the hot and dusty libraries of the metropolis. ... he had no ideas or sympathies in common with the physical philosopher. His attitude was doubtless due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted in an election, and he never visited the House of Commons, or the Tower, or Westminster Abbey.

De Morgan was always interested in odd numerical facts and, writing in 1864, he noted that he had the distinction of being $x$ years old in the year $x^{2}$ (He was 43 in 1849). Anyone born in 1980 can claim the same distinction in 2025.

For details of De Morgan's final years, see THIS LINK.

Five days after his death, on 23 March 1871, his funeral was held and he was buried at All Souls, Kensal Green, Kensington and Chelsea, London.