托马斯·霍布斯Thomas Hobbes

托马斯·霍布斯的父亲也叫霍布斯，是查尔顿和韦斯特波特的牧师，这两地靠近威尔特郡的马姆斯伯里。老霍布斯在13中被奥布里描述为：——

……伊丽莎白女王时代那些无知爵士约翰之一；只能读教会的祈祷文和布道文；不重视学问，因为不知其甘甜。

老霍布斯有一个哥哥Francis Hobbes，是一位富商，没有自己的家庭。本传记的主人公霍布斯有一个哥哥埃德蒙，比他大约两岁。霍布斯四岁时开始在韦斯特波特教堂上学。然而，他七岁时，父亲在教堂门口与另一位牧师发生争执。双方动了手，霍布斯的父亲跑掉了。此后母亲在他的教养中起了什么作用不清楚，但此后他肯定是由叔叔弗朗西斯抚养长大的。

从八岁起，霍布斯——此时已精通阅读和算术——进入马姆斯伯里的埃文先生学校，后来又进入韦斯特波特的罗伯特·拉蒂默私立学校。霍布斯在这所学校显露才华，十四岁离开时已是出色的希腊语和拉丁语学者，此前已将欧里庇得斯的Medea从希腊语翻译成拉丁语抑扬格诗。奥布里在13中告诉我们，霍布斯小时候有时顽皮，但有时也孤僻忧郁。在学校时常13：——

……他会躲到一个角落里，把功课背下来。

离开Robert Latimer的学校后，他于1603年进入牛津大学莫德林学院，在那里继续得到叔父Francis的经济支持。当时牛津的教学以对亚里士多德的研究为主导，而霍布斯很快发现自己的观点与所教授的内容[13]大相径庭：-

他不太喜欢逻辑学，但还是学了它，并认为自己是个不错的辩论者。他在那里的一大乐趣是去装订商的店铺，呆呆地盯着地图看。

他于1608年获得学士学位，并经莫德林学院院长James Hussey爵士推荐，成为William Cavendish（后来的第二代德文郡伯爵）的导师。在大约两年时间里，霍布斯在学术研究方面几乎没有什么作为，更多是作为Cavendish的同伴，而Cavendish只比他小一点。1610年，霍布斯与Cavendish一起进行欧洲之旅，他们访问了法国、德国和意大利。他在这次旅行中学会了法语和意大利语，但更重要的是，这重新激发了他对学习的渴望，他决定从事古典学研究。回来后，霍布斯再次开始学习希腊语和拉丁语。他从Cavendish的导师变成了他的秘书，由于职责很少，他有充足的时间投入学习。

1626年，父亲去世后，威廉·卡文迪什继承了德文郡伯爵爵位，但两年后威廉去世，霍布斯失去了一位朋友，也失去了秘书职位。威廉·卡文迪什的儿子只有十一岁，此时卡文迪什家族不再需要霍布斯的服务。

从1628年到1631年，霍布斯担任诺丁汉郡Gervase Clinton爵士之子的导师。在此期间，1629年，他出版了自己已经研究数年的修昔底德译本。到目前为止，我们还没有提到霍布斯对数学有任何兴趣，也许更令人惊讶的是，对哲学也没有特别的兴趣。事实上，霍布斯大约四十岁时才对数学着迷。尽管Aubrey对霍布斯初次接触数学的描述，像Aubrey的许多描述一样，有些夸张，但他在[13]中的描述仍然很值得记录：-

他四十岁时才看几何学；这是偶然发生的。他在一位绅士的书房里，欧几里得的《几何原本》摊开着，翻到的是第一卷第四十七命题。他读了那个命题。‘天哪，’他说，‘这不可能！’于是他读了它的证明，证明又把他引向某个引理；那个引理又把他引向另一个引理，他也读了。……最后，他通过证明确信了那个真理。这使他爱上了几何学。

1629年至1631年，他与新学生进行了第二次欧洲大陆之行。1631年，Cavendish家族再次请求他的服务，他从巴黎返回，成为第三代德文郡伯爵的导师，这一职位他从1631年担任到1642年。在此期间，他再次访问欧洲大陆，1634年至1637年在那里。在欧洲大陆，他遇到了伽利略、马兰·梅森、皮埃尔·伽桑狄和罗贝瓦尔，并对机械宇宙观产生了热情，开始构建他将一切与运动联系起来的哲学立场。事实上，他此时的观点似乎与当时最新的科学思想非常一致。1637年回到英国后，霍布斯致力于The Elements of Law, Natural and Politic，但当时并未出版。他在这部著作中描述了自己对感知的机械论方法如下：-

无论我们的感官让我们以为世界上存在什么偶性或性质，它们都不在那里，而只是现象和幻象；在我们之外的世界中真实存在的东西，是引起这些现象的运动。

1640年内战开始时，霍布斯担心自己的性命，尤其因为他是一位知名的保皇党人，于是他逃亡以保全性命。他从1640年起住在巴黎，在那里他再次与马兰·梅森的学者圈子取得了联系。在那里他写下了对勒内·笛卡儿的Meditations的反对意见，并于1642年出版了De Cive（《论公民》），其中包含了他关于教会与国家关系的思想。此后他研究光学，这是他最喜欢的主题之一。Malet [28]写道：-

霍布斯的光学图像理论[是]在他光学方面的巨著《光学初稿一瞥》（1646年）中发展起来的，并以节略版发表在《论人》（1658年）中。……霍布斯的视觉和图像理论使他能够将关于人的哲学建立在关于物体的哲学之上。此外，由于霍布斯光学工作的这一部分是最彻底几何化的，它大量揭示了数学在霍布斯哲学中的作用。

霍布斯于1647年出版了De Cive的新扩充版，然后三年后，即1650年，他较早的著作The Elements of Law, Natural and Politic未经他许可出版。它分两部分出现，即Humane Nature Ⓣ（《人性》）和De Corpore Politico Ⓣ（《论政治体》）。

霍布斯 在1646年至1648年间担任威尔士亲王的数学教师。他留在欧洲大陆直到1651年，那年他最著名的著作 Leviathan 出版，然后在那年年底，他回到了英国。事实上，他现在在政治光谱的各个方面都遇到了一些困难。在英国，随着查理一世去世，保皇党人似乎输掉了权力斗争。Leviathan 接近结尾的段落似乎表明 霍布斯 试图与英国政府和好，这激怒了保皇党人。事实上，在这些段落中，霍布斯 一直坚持他的观点，即一个人只有在该统治者能够提供保护时才表示效忠。霍布斯 还攻击了罗马天主教会，这使他在巴黎的处境变得相当难以维持。

霍布斯 的杰作 Leviathan 非常清晰地阐述了他的思想。他主张人们希望生活在和平与安全中，为了实现这一点，他们必须组织成共同体以寻求保护。由于共同体中总会有一些不可信任的人，人们必须建立一个政府，赋予其制定和执行保护共同体所需法律的权威。霍布斯 认为，这是人们行为的理性方式，因此道德行为是理性的。尽管 霍布斯 本人是基督徒，但这些论点被许多人视为消除了上帝作为道德准则给予者的必要性，因为 霍布斯 认为这仅凭理性即可得出。这部作品引起许多人攻击的另一个方面是 霍布斯 对大学体系的尖刻批评。

在此之前，霍布斯被许多人视为推动一种机械论科学方法，这与那些将组成皇家学会的人非常合拍。的确，他曾论证说，既然我们所知道和理解的一切只能通过我们的感官而来，而我们的感官能检测到的所有对象都是物质的，我们只能以物质的方式看待世界。他推动一种通过语言和数学来分析经验的方法，他声称这将导致对世界的完整机械论理解。数学的确定性将导致关于社会和人的正确且无可争辩的结论。他关于一切皆为物质的论点被视为否认非物质的灵魂和理智的存在。Seth Ward，牛津的萨维尔天文学讲席教授，写道：-

……他极大地损害了数学，以及证明这一名称本身，因为他把一些论述加诸其上，而这些论述远远达不到使一篇论述能够配得上这一声誉所需的那种证据和真理。

然而，在这个阶段，尽管霍布斯在数学方面发表的东西很少，但他确实被一些人视为与罗贝瓦尔和皮埃尔·德·费马齐名的领先数学家。

1655年，霍布斯出版了De Corpore Ⓣ（《论物体》），这是他哲学三部曲的一部分。他已经出版了De Cive Ⓣ（《论公民》）（1642年），第三部分De Homine Ⓣ（《论人》）将于1658年问世。De Corpore Ⓣ（《论物体》）包含了大量的数学材料；事实上，第12章到第20章完全致力于这个主题。霍布斯将数学视为知识的重要组成部分，但他也认为自己的唯物主义方法正在彻底改变这门学科，并在这部著作中着手改革数学。他的方法确实始终是唯物主义的，否认抽象观念；对霍布斯来说，数学是对量的研究，而量是三维物体的度量。他在De Corpore Ⓣ（《论物体》）中对点的定义（与欧几里得的定义完全不同）如下：-

如果一个被移动的物体的大小（尽管它必定总是有一定大小）被认为是无，那么它所经过的路径称为线，它所经过的空间称为长度，而物体本身称为点。正是在这个意义上，地球通常被称为一个点，而其周年公转的路径被称为黄道线。

因此，线是移动点的路径，面是移动线的路径，体是移动面的结果。然后他继续研究比和角，然后是加速度、抛射体以及伽利略的思想，接着研究不可分量的思想和博纳文图拉·卡瓦列里的思想，螺线的求长，最后是化圆为方。公平地说，霍布斯的许多数学思想是从伽利略对力学和运动的研究中推广而来的。博纳文图拉·卡瓦列里提出的新的不可分量方法被霍布斯接受，但他拒绝了约翰·沃利斯在Arithmetica infinitorum Ⓣ（《无穷小算术》）中给出的版本。

Jesseph关于霍布斯化圆为方的尝试写道[5]：-

……很明显，他希望在学术界确立卓越地位，很大程度上是基于对化圆为方问题的解决。

霍布斯最初计划写De CorporeⓉ（《论物体》）时并没有这个结果，后来才加上去，因此它与周围的内容并不真正契合。然而，在De CorporeⓉ（《论物体》）完成之前，霍布斯的朋友们指出了他化圆为方论证中的一个错误。霍布斯没有删去这个“证明”，而是把它改名为“从虚假的假设得出虚假的求积”。随后他又添加了第二个证明，但很快将其改为仅声称这是“一个近似求积”。最后他尝试了第三个精确证明，但当书正在印刷时，他意识到这个证明当然也是错的。他不得不保留这个错误的断言，但在该章末尾加上了：-

……读者应当把那些据说能精确求得圆之维度的事物……视为只是或然地说出的。

当约翰·沃利斯攻击霍布斯的思想时，他对这个说法嗤之以鼻。尽管霍布斯并不相信De CorporeⓉ（《论物体》）中的那些“证明”证明了该结果，但他在接下来的25年里仍陆续发表了若干化圆为方的“证明”，并且他相信这些证明是正确的。

约翰·沃利斯攻击了霍布斯De CorporeⓉ（《论物体》）中的全部数学工作，两人之间由此爆发了一场持续25年的激烈争论。对霍布斯来说，数学就是几何，而且仅仅是几何，他把约翰·沃利斯的Algebra描述为：-

……一堆符号的痂［把书页弄得面目全非］，就像有只母鸡在那里刨过一样。

霍布斯声称代数符号可以表示不同的东西，如线、面或体，因此在数学证明中不可靠。霍布斯针对约翰·沃利斯以及De CorporeⓉ（《论物体》）的其他人的攻击，于1656年出版了Six Lessons to the Professors of Mathematics in the University of Oxford。

1660年，霍布斯攻击了数学分析的“新”方法。在Dialogus Physicus, sive de Natura AerisⓉ（关于物理或空气性质的对话）（1661年）中，他攻击了罗伯特·波义耳以及那些建立皇家学会的人，有趣的是，该学会从未选举霍布斯为会士（很可能是因为他被视为无神论者，所以不可能入选）。约翰·沃利斯以有力的数学论据进行了回复，但也提出了不忠的指控，这是不公平的。霍布斯以Mr. Hobbes Considered in His Loyalty, Religion, Reputation, and Manners（1662年）结束了关于不忠的争论。当霍布斯的道德受到攻击时，他能够赢得争论，但在数学方面，约翰·沃利斯明显占上风，对数学的理解远比霍布斯深刻。

多年来，霍布斯试图解决若干悬而未决的数学问题。Jesseph在[22]中研究了：-

……霍布斯为解决十七世纪三个重要数学争议所做的尝试：关于解析几何地位的争论、关于比的本质的争论，以及曲线与切线之间“接触角”的问题。

霍布斯作为哲学家备受推崇，他的数学却基本上被人嘲笑。然而，有些人从中看到的不仅仅是错误。奥古斯塔斯·德摩根写道，霍布斯：-

……并非如有时人们所设想的那样在几何学上是个无知者。他的著作尽管在许多事情上是错误的，却包含了对原则性问题的敏锐评论。

Grant在[21]中评价了霍布斯的数学贡献，并得出结论说他是：-

……一位按这个词最初也最好的意义来说的数学业余爱好者，并且通过他作为他人成功的一种微小刺激因素的角色，他在数学史上理应占有一个不大的位置。

霍布斯 至死都在为自己的数学著作辩护。他的错误被如此清楚地揭示出来，以至于到1670年，基本上所有人都认为他是个数学文盲，然而他仍然写文章为自己辩护，尽管是否还有人继续读这些文章值得怀疑。让我们以 霍布斯 在生命接近终点时写下的、关于他自认为在数学上取得了什么成就的总结来结束。霍布斯 用第三人称写自己（例如见 [5]）：-

在数学中，他纠正了几何学的一些原理。他解决了一些最困难的问题，而这些问题自几何学诞生之初以来，最伟大的几何学家们勤勉细察也一直徒劳无功地寻求；即这些：

1. 展示一条等于圆周长度的线段，以及一个等于圆面积的平方，并通过多种方法实现。
2. 按给定比例分割一个角。
3. 求立方体与球体的比。
4. 求两条给定线段之间的任意数量的比例中项。
5. 描述具有任意边数的正多边形。
6. 求圆象限的重心。
7. 求所有类型抛物线的重心。

他是第一个构造并证明这些的人，此外还有许多其他东西（因为它们将出现在他的著作中，所以不那么重要），我略过不提。

他去世时91岁，在那个时代这是很了不起的年龄。87岁时，他完成了将《伊利亚特》和《奥德赛》译成英文韵文的工作，并离开了居住多年的伦敦，与他一生关系如此密切的卡文迪什家族一起度过了最后几年。91岁时，就在去世前不久，他还在写另一本关于化圆为方的书。献词中有这样一句话：-

于是，在我用不同方法对这个问题给予了足够关注之后——这些方法不被几何学教授们所理解——我加上了这个最新的方法。

据记载，他最后的话是：-

我即将进行最后一次航行，一次进入黑暗的巨大飞跃。

让我们以最后的思考结束这本传记。如果霍布斯的数学毫无价值，为什么即使在过去几年里也有如此多的努力被投入到它上面（如参考文献所示）。毫无疑问，霍布斯的数学是错误的，但奇怪的是，这似乎并没有使它变得毫无价值！作为哲学家，他是一位领军人物，对政治思想产生了重大影响。

Thomas Hobbes's father, also named Thomas Hobbes, was the vicar of Charlton and Westport, close to Malmesbury in Wiltshire. Thomas Hobbes senior was described by Aubrey in [13] as:-

... one of the ignorant Sir Johns of Queen Elizabeth's time; could only read the prayers of the church and the homilies; and valued not learning, as not knowing the sweetness of it.

Thomas Hobbes senior had an older brother, Francis Hobbes, who was a wealthy merchant with no family of his own. Thomas Hobbes, the subject of this biography, had one brother Edmund who was about two years older than he him. Thomas began his schooling in Westport Church when he was four years old. However, when he was seven years old, his father had an argument with another vicar at the door of his church. Blows were exchanged and Hobbes' father ran off. It is unclear what role his mother played in his upbringing after that, but he was certainly brought up by his uncle Francis after this.

From age eight Hobbes, who was by this time proficient at reading and arithmetic, attended Mr Evan's school in Malmesbury, then later Robert Latimer's private school in Westport. Hobbes showed his brilliance at this school and was an outstanding Greek and Latin scholar by the time he left this school at age fourteen, having already translated Euripides' Medea from Greek into Latin iambics. Aubrey in [13] tells us that as a young boy Hobbes was sometimes playful, but also sometimes withdrawn and melancholy. Often at school [13]:-

... he would get himself into a corner, and learn his lesson by heart.

After leaving Robert Latimer's school, he entered Magdalen Hall, Oxford in 1603 where he continued to be supported financially by his uncle Francis. At that time the teaching at Oxford was dominated by a study of Aristotle and Hobbes soon found that his opinions differed sharply from what was being taught [13]:-

He did not much care for logic, yet he learned it, and thought himself a good disputant. He took great delight there to go to the bookbinders' shops and lie gaping on maps.

He graduated with a B.A. in 1608 and on the recommendation of Sir James Hussey, Principal of Magdalen Hall, he became the tutor of William Cavendish, later the Second Earl of Devonshire. For around two years Hobbes did little in the way of academic studies, being more of a companion to Cavendish who was only a little younger that he was. In 1610 Hobbes went with Cavendish on a European tour and they visited France, Germany, and Italy. He learnt French and Italian on this trip, but more importantly, it reinvigorated his desire for learning and he decided that he would pursue a study of classics. On his return Hobbes took up studying Greek and Latin again. He had progressed from being a tutor to Cavendish to being his secretary and having few duties he had plenty of time to devote to his studies.

In 1626, on the death of his father, William Cavendish inherited the title the Earl of Devonshire, but two years later William died and Hobbes lost a friend as well as his secretarial post. William Cavendish's son was only eleven years old and Hobbes' services were no longer required by the Cavendish family at this time.

Hobbes was tutor to the son of Sir Gervase Clinton of Nottinghamshire, from 1628 to 1631. During this period, in 1629, he published his translation of Thucydides which he had been working on for several years. So far we have not mentioned any interest by Hobbes in mathematics, and perhaps even more surprisingly no particular interest in philosophy. In fact Hobbes was about forty years old before he became fascinated by mathematics. Although Aubrey's description of Hobbes encountering mathematics for the first time is, like so much of Aubrey, rather overdone, nevertheless his description in [13] is well worth recording:-

He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman's library Euclid's Elements lay open, and 'twas the forty-seventh proposition in the first book. He read the proposition. 'By God,' said he, 'this is impossible!' So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. ... at last he was demonstratively convinced of that truth. This made him in love with geometry.

He undertook a second trip to the continent from 1629 to 1631 with his new pupil. In 1631 the Cavendish family requested his services again and he returned from Paris to become tutor to the third Earl of Devonshire, a position he held from 1631 to 1642. During this time he again visited the continent, being there from 1634 to 1637. On the continent he met Galileo, Mersenne, Gassendi and Roberval and became enthusiastic about the mechanical universe and began building his philosophical position relating everything to motion. In fact his views at this time appeared to be very much in line with the latest scientific ideas of the period. Back in England in 1637 Hobbes worked on The Elements of Law, Natural and Politic which was not published at the time. He described his mechanistic approach to perception in this work as follows:-

Whatsoever accidents or qualities our senses make us think there be in the world, they be not there, but are seemings and apparitions only; the things that really are in the world without us, are those motions by which these seemings are caused.

When the Civil War began in 1640 Hobbes feared for his life, especially as he was a well known Royalist, and he fled to save his life. He lived in Paris from 1640 where again he made contact with Mersenne's circle of scholars. There he wrote his objections to Descartes' Meditations and he published De Cive (Concerning Citizenship) in 1642 which contained his ideas on the relation between the church and the state. After this he worked on optics, which was one of his favourite topics. Malet [28] writes:-

Hobbes's theory of optical images [was] developed in his optical magnum opus "A minute of first draught of the optiques" (1646) and published in abridged version in "De homine" (1658). ... Hobbes's theory of vision and images serves him to ground his philosophy of man on his philosophy of body. Furthermore, since this part of Hobbes's work on optics is the most thoroughly geometrical, it reveals a good deal about the role of mathematics in Hobbes's philosophy.

Hobbes published a new expanded edition of De Cive in 1647, then three years later, in 1650, his earlier work The Elements of Law, Natural and Politic was published without his permission. It appeared in two parts as Humane Nature Ⓣ and De Corpore Politico Ⓣ.

Hobbes was the mathematics tutor of the Prince of Wales between 1646 and 1648. He remained on the continent until 1651, the year his most famous work Leviathan was published then, late in that year, he returned to England. In fact he was now in some difficulties with all sides of the political spectrum. In England the Royalists, with Charles I dead, seemed to have lost their struggle for power. Passages near the end of the Leviathan appeared to indicate that Hobbes was trying to make his peace with the English government, which angered the Royalists. In fact in these passages Hobbes was remaining consistent with his view that one showed allegiance to a ruler only so long as that ruler could provide protection. Hobbes had also attacked the Roman Catholic Church which made his position in Paris pretty untenable.

Hobbes' masterpiece Leviathan set out his ideas with great clarity. He argued that people want to live in peace and security and to attain this they must organise themselves into communities for protection. Since there will always be some in the community who cannot be trusted, people must set up a government with their authority to make and enforce laws necessary to protect the community. It is, Hobbes argues, the rational way for people to behave so moral behaviour is rational. Although Hobbes was himself a Christian, these arguments were seen as many as removing the need for God as the giver of moral code, for Hobbes argues that it follows by reason alone. Another aspect of the work which caused many to attack it was Hobbes' vitriolic arguments against the university system.

Before this Hobbes had been seen by many as promoting a mechanistic scientific approach which was much in tune with those who would form the Royal Society. Indeed he had argued that since what we know and understand only comes through our senses and all objects that our senses can detect are material, we can only view the world in a material way. He promoted an approach through language and mathematics to analyse experience which he claimed would lead to a complete mechanistic understanding of the world. The certainty of mathematics would lead to correct and indisputable conclusions about society and about man. His argument that all was material was seen as denying the existence of the immaterialistic soul and intellect. Seth Ward, the Savilian Professor of Astronomy at Oxford, wrote:-

... he hath much injured the mathematics, and the very name of demonstration, by bestowing upon it some of his discourses, which are exceedingly short of that evidence and truth which is required to make a discourse able to bear that reputation.

At this stage, however, although Hobbes had published little in the way of mathematics, he certainly was considered by some as a leading mathematician on a par with Roberval and Fermat.

In 1655 Hobbes published De Corpore Ⓣ which, was one part of his trilogy of philosophy. He had already published De Cive Ⓣ (1642) and the third part, De Homine Ⓣ, would appear in 1658. De Corpore Ⓣ contained a large amount of mathematical material; in fact Chapters 12 to 20 are devoted entirely to the topic. Hobbes saw mathematics as an essential part of knowledge, but he also saw his own materialistic approach as revolutionising the subject and he set out to reform mathematics in this work. His approach is certainly consistently materialistic, denying abstract ideas; for Hobbes mathematics is the study of quantity, and quantities are the measures of 3-dimensional bodies. His definition of a point in De Corpore Ⓣ (which totally differs from that of Euclid) is as follows:-

If the magnitude of a body which is moved (although it must always have some) is considered to be none, the path by which it travels is called a line, and the space it travels along a length, and the body itself is called a point. This is the sense in which the earth is usually called a point and the path of its annual revolution the ecliptic line.

Lines, therefore, are the paths of moving points, surfaces are the paths of moving lines, volumes are the result of moving surfaces. He then proceeded to study ratios and angles, then acceleration, projectiles and the ideas of Galileo followed by a study of indivisibles and the ideas of Cavalieri, the rectification of the spiral, and finally squaring the circle. It is fair to say that much of Hobbes' mathematical ideas are generalised from Galileo's study of mechanics and of motion. The new method of indivisibles, as put forward by Cavalieri, was accepted by Hobbes but he rejected Wallis's version as given in Arithmetica infinitorum Ⓣ.

Jesseph writes of Hobbes' attempt to square the circle [5]:-

... it is clear that he hoped to assert preeminence in the learned world largely on the basis of the solution of the problem of squaring the circle.

Hobbes had originally planned De Corpore Ⓣ without this result and, having added it late on, it did not really fit with the material surrounding it. Before De Corpore Ⓣ reached completion, however, Hobbes' friends pointed out an error in his squaring the circle argument. Hobbes did not remove the "proof" but renamed it "From a false hypothesis, a false quadrature". He then added a second proof which he quickly changed to only claim it was "an approximate quadrature". Finally he attempted a third exact proof but when the book was being printed he realised that it too, of course, was wrong. He had to leave the incorrect claim but added at the end of the chapter:-

... the reader should take those things that are said to be found exactly of the dimension of the circle ... as instead said problematically.

This was a phrase that Wallis would pour scorn on when he attacked Hobbes' ideas. Although Hobbes did not believe that the "proofs" in De Corpore Ⓣ proved the result, he would go on to publish several "proofs" of squaring the circle over the next 25 years which he did believe to be correct.

Wallis attacked the whole of Hobbes' mathematical work of De Corpore Ⓣ and a vigorous argument between the two arose which lasted for 25 years. To Hobbes mathematics was geometry and only geometry, and Wallis's Algebra he described as:-

... a scab of symbols [which disfigured the page] as if a hen had been scraping there.

Hobbes claimed that the algebraic symbols could denote different things such as lines, surfaces or volumes, and therefore were unreliable in mathematical proofs. Hobbes responded to the attack by Wallis and others of De Corpore Ⓣ by publishing Six Lessons to the Professors of Mathematics in the University of Oxford in 1656.

In 1660 Hobbes attacked the 'new' methods of mathematical analysis. In Dialogus Physicus, sive de Natura Aeris Ⓣ (1661) he attacked Boyle and those setting up the Royal Society which, as a matter of interest, never elected Hobbes as a Fellow (it is probably that since he was perceived as an atheist entry would have been impossible). Wallis replied with telling mathematical arguments, but also with unfair charges of disloyalty. Hobbes ended the argument about disloyalty with Mr. Hobbes Considered in His Loyalty, Religion, Reputation, and Manners (1662). Hobbes could win arguments when his morality was attacked, but when it came to mathematics Wallis had a clear upper hand understanding mathematics far more deeply than Hobbes.

Over the years Hobbes attempted to solve a number of outstanding mathematical problems. Jesseph, in [22], studies:-

... Hobbes's attempts to resolve three important mathematical controversies of the seventeenth century: the debates over the status of analytic geometry, disputes over the nature of ratios, and the problem of the 'angle of contact' between a curve and tangent.

Although Hobbes is highly regarded as a philosopher, his mathematics has been essentially laughed at. However some have seen more in it than just errors. De Morgan wrote that Hobbes:-

... was not the ignoramus in geometry that he is sometimes supposed. His writings, erroneous as they are in many things, contain acute remarks on points of principle.

Grant, in [21], evaluates Hobbes' mathematical contributions and concludes that he was:-

... an amateur of mathematics in the original and best sense of the word, and through his role as a minor stimulant of others' success he merits a modest place in its annals.

Hobbes defended his mathematical works to the end of his life. His errors were demonstrated so clearly that by 1670 essentially everyone considered him a mathematical illiterate, yet still he wrote articles in his defence even though it is doubtful whether anyone continued to read them. Let us end with the summary of what Hobbes believed that he had achieved in mathematics, written near the end of his life. Hobbes writes about himself in the third person (see for example [5]):-

In mathematics, he corrected some principles of geometry. he solved some most difficult problems, which had been sought in vain by the diligent scrutiny of the greatest geometers since the very beginnings of geometry; namely these:

1. To exhibit a line equal to the arc of a circle, and a square equal to the area of a circle, and this by various methods.
2. To divide an angle in a given ratio.
3. To find the ratio of a cube to a sphere.
4. To find any number of mean proportionals between two given lines.
5. To describe a regular polygon with any number of sides.
6. To find the centre of gravity of the quadrant of a circle.
7. To find the centres of gravity of all types of parabolas.

He was the first to construct and demonstrate these, and many other things besides, which (because they will appear in his writings are less important) I pass over.

He was 91 years of age when he died, a remarkable age for someone in that period. At age 87 he completed translating the Iliad and the Odyssey into English verse and left London, where he had lived for many years, and spent his final years with the Cavendish family with whom he had been so closely connected throughout his life. At age 91, shortly before his death, he was working on yet another book on squaring the circle. The dedication contains the sentence:-

And so, after I had given sufficient attention to the problem by different methods, which were not understood by the professors of geometry, I added this newest one.

His final words are reported to have been:-

I am about to take my last voyage, a great leap in the dark.

Let us end this biography with a final thought. If Hobbes' mathematics was worthless why has so much effort been expounded on it even in the last few years (as the references show). There is no doubt that Hobbes' mathematics is wrong, but strangely, that does not seem to make it worthless! As a philosopher he was a leading figure, having a major influence on political thought.