埃伦弗里德·瓦尔特·冯·切恩豪斯Ehrenfried Walther von Tschirnhaus

埃伦弗里德·瓦尔特·冯·切恩豪斯（或Tschirnhausen）出生于德国的Kieslingswalde，但自1945年以来，该镇被称为Sławnikowice，位于波兰西部。他是父母Christoph 冯·切恩豪斯和Elisabeth Eleonore Freiin Achyll von 斯特林的最小的儿子和第七个孩子。Christoph是撒克逊贵族的土地所有者，而Elisabeth是德国和苏格兰血统，与数学天赋的斯特林家族有远亲关系。当冯·切恩豪斯六岁时，他的母亲去世了，但他由一位慈爱的继母抚养长大。他的教育一直由私人教师进行，直到他十五岁，这为他的人生提供了极好的广泛教育开端。1666年，他进入位于Görlitz的文理中学，在那里度过了两年准备大学入学。在这个阶段，他对数学有着浓厚的兴趣，并私下学习这门学科，以超越在学校学到的材料。

他于1668年秋进入莱顿大学，在那里学习数学、哲学、物理学和医学。他还在1669年6月8日在莱顿注册了法学院，但似乎对那个学科没有太大兴趣。他了解到医学的最新进展，例如哈维关于血液循环的理论。他还从Pieter van 弗兰斯·范斯霍滕（著名数学家弗兰斯·范斯霍滕的同父异母兄弟）那里接受私人课程，后者向他介绍了勒内·笛卡儿的数学和哲学。1672年，荷兰和法国之间爆发战争，冯·切恩豪斯加入了学生队伍。他没有参加实际战斗，但尽管如此，这意味着他的学业中断了十八个月。冯·切恩豪斯于1674年开始欧洲之旅。从Kieslingswalde出发，他首先回到莱顿，在那里他的校友Pieter van Gent将他介绍给Baruch de Spinoza。冯·切恩豪斯成为Spinoza主持的辩论俱乐部的常客，这使他渴望创办一个科学院，后来他试图建立。我们顺便注意到，他不仅是van Gent的终身朋友：他还是他的赞助人，以至于后来他支付他作为通讯员、编辑和翻译的服务费用。带着Spinoza的推荐信，冯·切恩豪斯于1675年5月访问英国，在那里他会见了皇家学会的秘书Oldenburg。Oldenburg将他介绍给罗伯特·波义耳、德尼·帕潘和艾萨克·牛顿等人。他还在伦敦会见了约翰·柯林斯，在牛津会见了约翰·沃利斯。他向约翰·柯林斯和约翰·沃利斯展示了他解方程的方法，但这些结果后来证明是已知结果的特例。带着Oldenburg的推荐信，他于1675年秋前往巴黎，在那里停留了一段时间，会见了哥特弗里德·威廉·莱布尼茨和克里斯蒂安·惠更斯。Nadler [3]引用了G H Schuller在1675年秋写给Spinoza的一封信，信中描述了冯·切恩豪斯在巴黎会见哥特弗里德·威廉·莱布尼茨的情形：-

[冯·切恩豪斯]在[巴黎]遇到了一位名叫哥特弗里德·威廉·莱布尼茨的人，学识非凡，精通各门科学，并且摆脱了常见的神学偏见。[冯·切恩豪斯]与他建立了密切的友谊，基于这样一个事实：他和他一样，正在研究完善理智的问题，而且他认为没有什么比这更好或更重要。在伦理学方面，他说，哥特弗里德·威廉·莱布尼茨最为精通，并且完全根据理性的指示说话，不受情感影响。他补充说，在物理学，特别是在关于上帝和灵魂的形而上学研究中，他最为精通……

哥特弗里德·威廉·莱布尼茨安排冯·切恩豪斯研究勒内·笛卡儿未发表的论文，他还获得了布莱兹·帕斯卡和罗贝瓦尔未发表的论文。在研究这些之后，冯·切恩豪斯在给G H Schuller、Pieter van Gent和Spinoza的信中报告了他的发现。他会见了François Villette，并观察了他正在进行的关于燃烧镜和矿物熔化的众多实验。在巴黎期间，冯·切恩豪斯教了Jean-Baptiste Colbert的一个儿子，但由于冯·切恩豪斯不懂法语，课程不得不用拉丁语进行。1676年11月，他离开巴黎，陪同西里西亚的Nimpsch伯爵。他们首先前往里昂，在那里冯·切恩豪斯再次会见了Villette，并与他一起进行了燃烧镜实验。然后他前往都灵、米兰、威尼斯、博洛尼亚和罗马。他所到之处都与顶尖科学家接触，包括罗马的Athanasius Kircher和Alfonso 吉奥万尼·阿方索·博雷利。他的旅行继续，访问了那不勒斯、西西里、米兰和日内瓦，然后于1679年返回巴黎、海牙（在那里他拜访了克里斯蒂安·惠更斯）和汉诺威（在那里他拜访了哥特弗里德·威廉·莱布尼茨）。在这次长途旅行中，冯·切恩豪斯继续通过信件向哥特弗里德·威廉·莱布尼茨报告他的观察和发现，并收到有用的回复。例如，1678年4月30日，冯·切恩豪斯从罗马给哥特弗里德·威廉·莱布尼茨写了一封长信（见[10]）。在信中，他讨论了几个数学问题，包括高次方程的求解。他提出代数学是一门广泛的学科，其中包含组合数学作为一部分。他写道：-

许多人错误地认为组合数学艺术是一门独立的科学，必须在代数学和其他科学之前掌握。事实上，有些人认为组合数学艺术中的内容比通常称为代数学的艺术更多；换句话说，女儿知道的比母亲多。但仅从幂的构成来看，即使没有别的，也显然表明组合数学艺术是通过代数学来掌握的。

哥特弗里德·威廉·莱布尼茨 回复道：-

……你的话无疑针对我，因为如你所说，以这种方式思考的“许多人”，我相信，除我之外寥寥无几。然而，我相信你的观点是正确的，因为你似乎没有理解我。因为如果你认为组合术是寻找变差数的科学，我坦然承认它从属于数的科学，因而从属于代数，因为数的科学也从属于代数。……但对我来说，组合术实际上远非如此，而是形式或相似与不相似的科学，而代数则是量或相等与不等的科学。

在他的信中，哥特弗里德·威廉·莱布尼茨 还批评了 冯·切恩豪斯 对代数方程的解法。他还阐述了他所发展的微积分的一些基本原理。

1679年结束漫长的旅行后，冯·切恩豪斯回到家乡基斯林斯瓦尔德住了一段时间。他在机械师约翰·霍夫曼的协助下，从事圆形和抛物面镜的制造。这些镜子使他能通过聚焦阳光获得高温。然而，他很快又踏上旅途，于1680年经荷兰和比利时前往巴黎。1682年他第三次访问巴黎，并于7月22日当选为Académie des Sciences院士。他曾希望获得养老金，以便在经济上自由地继续科学研究，但未能如愿。同样在1682年，他在基斯林斯瓦尔德与伊丽莎白·埃莱奥诺雷·冯·莱斯特结婚。她是萨克森选帝侯宫廷一位重要成员的千金。1684年，冯·切恩豪斯的父亲去世，将基斯林斯瓦尔德的家族庄园留给了他。冯·切恩豪斯的妻子接管了庄园的管理，以便他有时间继续科学研究。

冯·切恩豪斯 研究方程的解法和曲线。他发现了一种变换，当应用于一个 $n$ 次方程时，得到一个 $n$ 次方程，其中不含 $x^{n-1}$ 和 $x^{n-2}$ 项。我们上面已经指出，他已经与 哥特弗里德·威廉·莱布尼茨 讨论过解方程的方法，后者指出了困难。尽管如此，冯·切恩豪斯 于1683年在 Acta Eruditorum 上发表了他的变换，并在这篇文章中展示了如何用它来解一般三次方程。然而，他认为该方法可以解任意次方程的信念是错误的，正如 哥特弗里德·威廉·莱布尼茨 已经向他指出的那样。他还在1682年研究了回光线，即从点光源发出的光线经给定曲线反射后的 包络。他在曲线方面的工作被人们铭记，因为一种正弦螺线以他的名字命名。大约在这个时候，冯·切恩豪斯 正计划写一部阐述他哲学的重要著作。1682年，他寄了一份他打算 克里斯蒂安·惠更斯 的方案，但过了六年这部作品才完成。1686年，他出版了 Medicina corporis Ⓣ（物体的哲学），然后在次年出版了 Medicina mentis Ⓣ（心灵的哲学）。这两部作品被合为一卷，也于1687年出版，然后1695年出版了第二版，标题为 Medicina mentis sive artis inveniendi praecepta generali Ⓣ（发现心灵哲学或艺术的通用规则）。Van Peursen 写道[14]：-

他沿着理性主义思想家如 勒内·笛卡儿、尼古拉斯‧马勒伯朗士 和 Spinoza 已经勾勒的路线前进。他的独创性在于努力纠正这些哲学。他通过将他的整个哲学置于发明的视角下来做到这一点。

冯·切恩豪斯 论证道[14]：-

……一个人可以在不知道其实际运作方式的情况下进行智力及其他操作。冯·切恩豪斯经常举我们使用双手却对其生理结构一无所知的例子。因此，我们可以钦佩一位钟表匠的手工能力和技巧，而他对自己的双手如何运作却一无所知。冯·切恩豪斯哲学的整体方法正是基于这一观念，使读者能够以最自然甚至天真的方式实践ars inveniendi。

1700年，冯·切恩豪斯出版了Gründliche Anleitung zu nützlichen WissenschaftenⓉ（有用科学详尽指南），该书受到哥特弗里德·威廉·莱布尼茨的赞扬，并极大地影响了Christian Wolff。

Hofmann [1] 写到冯·切恩豪斯对数学的浓厚兴趣：-

冯·切恩豪斯在寻找算法的过程中耗尽了他的数学才能。由于缺乏对数学命题之间更深刻关系的洞察，他过于轻易地根据所获得的特定结果断言一般关系的存在。此外，他不愿直接接受其他数学家的建议，尽管他后来会将这些建议采纳为自己的发明并如此发表。这种策略导致了与哥特弗里德·威廉·莱布尼茨、克里斯蒂安·惠更斯、菲利普·德拉伊尔、雅各布·伯努利和约翰·伯努利的激烈争论，并最终使他失去了科学声誉。

本引文提到的与哥特弗里德·威廉·莱布尼茨的争论发生在1682-84年，涉及代数曲线代数求积的可能性。在Medicina mentisⓉ（心灵哲学）出版后，Fatio de Duiller正确地指出，冯·切恩豪斯在书中提出的求曲线切线的方法是不正确的。这场争论也在1687-89年间持续了几年。

除了在哲学和数学方面的工作外，冯·切恩豪斯还是一位科学家，除其他事项外，他在1680年代实验用黏土混合可熔岩石制作瓷器。他的许多实验涉及使用燃烧镜，借此他能够产生比以往更高的温度。1694年，他宣布在前一年冬天成功进行了生产瓷器的实验。到1696年，他正在与萨克森选帝侯奥古斯特二世讨论建造玻璃和瓷器工厂的事宜。选帝侯提出了条件，首先冯·切恩豪斯要找到[12]：-

……萨克森所有出产碧玉、玛瑙、紫晶和黄玉等宝石的地方。

这些地方将提供工厂所需的原材料，并指示出建厂的最佳地点。约1699年，玻璃工厂在德累斯顿和格吕克斯堡建立，其建造由冯·切恩豪斯监督。他在1701至1702年冬季前往荷兰，考察了代尔夫特等地的陶瓷工厂，然后才将自己的瓷器投入生产。从1702年起，他与约翰·弗里德里希·伯特格[12]合作：-

……一个有过化学经验和实验室技能的骗子兼惯犯……

研究生产瓷器的问题。冯·切恩豪斯的妻子伊丽莎白已于1692年去世，1704年2月他再婚，第二任妻子是伊丽莎白·冯·德·舒伦堡·祖·米尔巴赫。1706年瑞典入侵萨克森，迫使奥古斯特于9月签署《阿尔特兰施泰特条约》正式退位。这使冯·切恩豪斯的瓷器工厂陷入极大困境。然而战后，他被授予哈勒大学校长的职位，但仍留在其家族领地基斯林斯瓦尔德。各国政府为获取他的瓷器技术展开了激烈竞争，但冯·切恩豪斯将技术秘而不宣，最终在深陷债务中结束了一生。舍恩费尔德写道[12]：-

冯·切恩豪斯和伯特格于[1708]年10月烧制出第一件真正的无釉瓷器杯。该杯由在1350°C以上烧制的钙质瓷制成（这一温度可产生硬质瓷）。其不熔成分是来自科尔迪茨的高岭土，可熔成分是雪花石膏和硫酸钙。这一突破后仅数日，冯·切恩豪斯便因痢疾去世。“胜利！成功！”是他最后的遗言。

1710年，迈森的一家工厂开始生产他的瓷器，次年维也纳也有一家工厂投产，而冯·切恩豪斯瓷器的首次大规模销售于1713年在莱比锡博览会上进行。

Hofmann 对他的贡献作了如下评价 [1]：-

在大学期间，他缺乏一位善良、有经验而又严格的老师的指导，这样的老师本可以约束他过于旺盛的性情，缓和他对 勒内·笛卡儿 思想的过度热情，并使他养成更多的自我批评精神。即便如此，冯·切恩豪斯 的成就——常常是在条件不足的情况下取得的——远比他同时代大学科学教师的平均贡献更为重要。事实上，甚至他的错误也被证明是对其他科学家重要而有成果的激励。

虽然这些批评意见在针对 冯·切恩豪斯 的数学贡献时可能是公允的，但在考虑他的哲学贡献时可能就过于苛刻了，因为他在哲学上确实超越了 勒内·笛卡儿 的思想。

Ehrenfried Walter von Tschirnhaus (or Tschirnhausen) was born in Kieslingswalde in Germany, but since 1945 the town has been called Sławnikowice and has been in western Poland. He was the youngest son and seventh child of his parents Christoph von Tschirnhaus and Elisabeth Eleonore Freiin Achyll von Stirling. Christoph was a landowner from the Saxon nobility while Elisabeth was of German and Scottish origin, distantly related to the mathematically gifted Stirling family. When Tschirnhaus was six years old his mother died but he was brought up by a loving stepmother. His education was from private tutors until he was fifteen years old and this gave him an excellent broad educational start to life. In 1666 he entered the Gymnasium in Görlitz where he spent two years preparing for university entrance. He had a deep interest in mathematics at this stage and he took private lessons in the subject to take him beyond the material learnt at school.

He entered the University of Leiden in the autumn of 1668 and there he studied mathematics, philosophy, physics and medicine. He also matriculated in the law faculty in Leiden on 8 June 1669 but seems not to have had much of an interest in that topic. He learnt of the latest advances in medicine such as Harvey's theory concerning the circulation of blood. He also took private lessons from Pieter van Schooten (half-brother of the famous mathematician Frans van Schooten) who introduced him to both the mathematics and philosophy of Descartes. In 1672 war broke out between Holland and France and Tschirnhaus enlisted in the student force. He did not see active service but, nevertheless, it meant that he had an eighteen-month interruption to his studies. Tschirnhaus began a European tour in 1674. Setting out from Kieslingswalde he first returned to Leiden where his school friend Pieter van Gent introduced him to Baruch de Spinoza. Tschirnhaus became a regular participant in the debating club which Spinoza ran and this gave him the desire to run an Academy which he later tried to set up. We note in passing that he became more than a life-long friend to van Gent: he was also his patron to the extent that he later paid him for his services as a correspondent, editor and translator. With a letter of recommendation from Spinoza, Tschirnhaus visited England in May 1675 where he met Oldenburg the secretary of the Royal Society. Oldenburg introduced him to, among others, Robert Boyle, Denis Papin, and Isaac Newton. He also met John Collins in London and John Wallis in Oxford. He showed Collins and Wallis his methods for solving equations, but these turned out to be special cases of known results. With a letter of recommendation from Oldenburg, he went to Paris in the autumn of 1675 where he remained for a while after meeting Leibniz and Huygens. Nadler [3] quotes a letter from G H Schuller to Spinoza written in the autumn of 1675 which describes Tschirnhaus meeting Leibniz in Paris:-

[Tschirnhaus] has met [in Paris] a man named Leibniz of remarkable learning, most skilled in the various sciences and free from the common theological prejudices. [Tschirnhaus] has established a close friendship with him, based on the fact that like him he is working at the problem of the perfecting of the intellect, and indeed he considers there is nothing better or more important than this. In ethics, he says, Leibniz is most practised, and speaks solely from the dictates of reason uninfluenced by emotion. He adds that in physics and especially in metaphysical studies of God and the Soul he is most skilled ...

Leibniz arranged for Tschirnhaus to study the unpublished papers of Descartes and he also was given access to unpublished papers by Pascal and Roberval. After studying these, Tschirnhaus reported his findings in letters to G H Schuller, Pieter van Gent and Spinoza. He met François Villette and observed numerous experiments he was carrying out on burning mirrors and melting of minerals. While in Paris, Tschirnhaus taught one of Jean-Baptiste Colbert's sons but, as Tschirnhaus did not know French, the lessons had to be in Latin. In November 1676 he left Paris, accompanying Count Nimpsch of Silesia. They first travelled to Lyon where Tschirnhaus again met Villette and carried out experiments with him on burning mirrors. He then travelled to Turin, Milan, Venice, Bologna and Rome. Everywhere he went he made contact with the leading scientists including Athanasius Kircher and Alfonso Borelli in Rome. His travels continued with visits to Naples, Sicily, Milan, and Geneva before returning in 1679 to Paris, The Hague (where he visited Huygens) and Hanover (where he visited Leibniz). While making this long journey, Tschirnhaus continued to report his observations and discoveries to Leibniz by letter, receiving helpful replies. For example, on 30 April 1678 Tschirnhaus wrote a long letter to Leibniz from Rome (see [10]). In it he discussed several mathematical questions including the solution of higher equations. He proposed that algebra is a wide-ranging subject which contains combinatorics as a part. He wrote:-

Many people quite falsely believe that the art of combinatorics is a separate science, to be mastered before algebra and other sciences. Indeed, some people believe that there is more in the art of combinatorics than in the art commonly called algebra; in other words, that the daughter knows more than the mother. But it is certainly obvious, from the composition of powers alone if by nothing else, that the art of combinatorics is mastered through algebra.

Leibniz replied:-

... your words are undoubtedly aimed at me, for the 'many' who, as you say, think in this way are few, I believe, beside myself. However, I believe that your opinion is right because you do not seem to have understood me. For if you hold the art of combinatorics to be the science of finding the number of variations, I freely admit that it is subordinate to the science of numbers and consequently to algebra, since the science of numbers is also subordinate to algebra. ... But for me the art of combinatorics is in fact something far different, namely, the science of forms or of similarity and dissimilarity, while algebra is the science of magnitude or of equality and inequality.

In his letter Leibniz also criticises Tschirnhaus's solution of algebraic equations. He also sets out some of the fundamental principles of the calculus that he has developed.

After ending his long journey in 1679, Tschirnhaus lived for a time back in his home town of Kieslingswalde. He worked on the construction of circular and parabolic mirrors, aided by his mechanic Johann Hoffmann. These allowed him to obtain high temperatures by focussing sunlight. He was soon on his travels again, however, going to Paris via Holland and Belgium in 1680. He made a third visit to Paris in 1682, and on 22 July he was elected to the Académie des Sciences. He had hoped for a pension to give him financial freedom to continue his scientific studies, but none was forthcoming. Also in 1682 he married Elisabeth Eleonore von Lest in Kieslingswalde. She was the daughter of an important member of the court of the Elector of Saxony. In 1684 Tschirnhaus's father died leaving him the family estate at Kieslingswalde. Tschirnhaus's wife took over managing the estate to allow him the time to continue his scientific researches.

Tschirnhaus worked on the solution of equations and the study of curves. He discovered a transformation which, when applied to an equation of degree $n$, gave an equation of degree $n$ with no term in $x^{n-1}$ and $x^{n-2}$. We have indicated above that he had already discussed his methods for solving equations with Leibniz who had pointed out difficulties. Nevertheless Tschirnhaus published his transformation in Acta Eruditorum in 1683 and, in this article, showed how it could be used to solve the general cubic equation. However, his belief that the method would allow an equation of any degree to be solved is false as had already been pointed out to him by Leibniz. He also studied catacaustic curves in 1682, these being the envelope of light rays emitted from a point source after reflection from a given curve. His work on curves is remembered since a sinusoidal spiral is named after him. Around this time Tschirnhaus was planning to write a major work explaining his philosophy. In 1682 he sent a scheme of what he intended to Huygens but it was six more years before the work came to fruition. In 1686 he published Medicina corporis Ⓣ, then in the following year he published Medicina mentis Ⓣ. The two were put together as a single volume which also appeared in 1687, then a second edition was published in 1695 under the title Medicina mentis sive artis inveniendi praecepta generali Ⓣ. Van Peursen writes [14]:-

He drew lines already traced by rationalist thinkers such as Descartes, Malebranche, and Spinoza. His originality lies in the effort to correct these philosophies. He did this by placing his whole philosophy in the perspective of invention.

Tschirnhaus argues that [14]:-

... a person can perform intellectual and other operations without knowing how they actually work. Tschirnhaus frequently gives the example of the way in which we use our hands without any knowledge of their physiological structure. Thus we can admire the manual ability and skill of a watchmaker who does not know anything at all about the way in which his hands function. The whole approach of Tschirnhaus's philosophy is based on this idea, so that the readers can practice the ars inveniendi in a most natural and even naive way.

In 1700 Tschirnhaus published Gründliche Anleitung zu nützlichen Wissenschaften Ⓣ which was praised by Leibniz and greatly influenced Christian Wolff.

Hofmann [1] writes of Tschirnhaus's deep interest in mathematics:-

Tschirnhaus exhausted his mathematical talents in searching for algorithms. Lacking insight into the more profound relations among mathematical propositions, he was too ready to assert the existence of general relationships on the basis of particular results that he obtained. Further he was unwilling to accept suggestions directly from other mathematicians, although he would later adopt them as his own inventions and publish them as such. This tactic led to bitter controversies with Leibniz, Huygens, La Hire and Jacob Bernoulli and Johann Bernoulli, and it ultimately cost him his scientific reputation.

The dispute with Leibniz referred to in this quotation took place in 1682-84 and involved the possibility of the algebraic quadrature of algebraic curve. After the publication of Medicina mentis Ⓣ, Fatio de Duiller correctly claimed that a method presented by Tschirnhaus in the book to find tangents to curves was incorrect. This argument also went on for a couple of years during 1687-89.

As well as his work in philosophy and mathematics, Tschirnhaus was a scientist, and among other things, he experimented making porcelain from clay mixed with fusible rock in the 1680s. Much of his experimenting involved the use of burning mirrors with which he was able to generate higher temperatures than had previously been produced. In 1694 he announced that he had been successful in experiments to produce porcelain in the previous winter. By 1696 he was in discussions with the Saxon elector, Augustus II, concerning the building of glass and porcelain factories. The elector set out conditions that first Tschirnhaus find [12]:-

... all places in Saxony with deposits of the precious stones jasper, agate, amethyst, and topaz.

These would provide the raw materials needed for the factories and indicate the best places to site them. Glass factories were established in Dresden and Glücksburg around 1699 and their construction was supervised by Tschirnhaus. He made a trip to Holland in the winter of 1701-02, inspecting the ceramic factories in Delft and other places before putting his porcelain into production. From 1702 he worked with Johann Friedrich Böttger [12]:-

... a con man and a jailbird with chemical experience and laboratory skills ...

on the problem of producing porcelain. Tschirnhaus's wife Elisabeth had died in 1692 and in February 1704 he remarried, his second wife being Elisabeth von der Schulenburg zu Mühlbach. In 1706 Sweden invaded Saxony, forced Augustus to formally abdicate by signing the Treaty of Altranstädt in September. This put Tschirnhaus in considerable difficulty regarding his porcelain factories. However after the war he was offered the position of Chancellor at the University of Halle but remained on his family estate of Kieslingswalde. There was great competition from governments to obtain his porcelain techniques but Tschirnhaus kept them to himself and ended his life deeply in debt. Schönfeld writes [12]:-

Tschirnhaus and Bottger fired the first cup of true unglazed porcelain in October [1708]. The cup consisted of a calcareous porcelain burned above 1,350°C (a temperature that produces a hard-paste porcelain). Its infusible ingredient was kaolin clay from Colditz, and its fusible ingredients were alabaster and calcium sulphate. Only days after this breakthrough, Tschirnhaus succumbed to dysentery and died. "Triumph! Victory!" were his last words.

A factory at Meissen started production of his porcelain in 1710, one in Vienna in the following year, and the first sales of any consequence of Tschirnhaus's porcelain took place at the Leipzig Fair in 1713.

Hofmann gives this assessment of his contributions [1]:-

During his university years he lacked the guidance of a kind, experienced, yet strict teacher, who could have restrained his exuberant temperament, moderate his excessive enthusiasm for Descartes' ideas, and instilled in him a greater measure of self-criticism. Even so, Tschirnhaus's achievements - often accomplished with insufficient means - were far more significant than the average contribution made by university teachers of science during his lifetime. Indeed, even his errors proved to be important and fruitful stimuli for other scientists.

Although these critical comments are probably fair when directed towards Tschirnhaus's mathematical contributions, they are probably harsh when his philosophical contributions are considered where he did go beyond the ideas of Descartes.