赫尔曼·格拉斯曼Hermann Grassmann

赫尔曼·格拉斯曼的父亲是Justus Günter 格拉斯曼，母亲是Johanne Luise Friederike Medenwald，她是来自Klein-Schönfeld的一位牧师的女儿。Justus曾被任命为牧师，但他在斯德丁的文理中学担任数学和物理教师。他是一位优秀的学者，写了几本关于物理和数学的教科书，还进行了晶体学研究。Johanne和Justus有十二个孩子，雅各布·赫尔曼是他们的第三个孩子。赫尔曼的兄弟Robert也成为了一名数学家，两人合作了许多项目。

当赫尔曼年幼时，他由母亲教导，她是一位受过良好教育的女性。然后他上了一所私立学校，之后进入斯德丁的文理中学，他父亲在那里教书。这个档案中的大多数数学家从小就给老师留下深刻印象，但令人惊讶的是，尽管在一个重视教育的家庭中拥有极好的教育机会，赫尔曼在文理中学的头几年并没有表现出色。他的父亲觉得他应该以园艺师或工匠等体力工作为目标。赫尔曼确实在音乐中找到乐趣，并学会了弹钢琴。随着他在学校的进步，他确实慢慢提高了，到十八岁参加中学毕业考试时，他在学校排名第二。在证明了自己至少是一位非常称职的学者后，赫尔曼决定学习神学，他于1827年与他的长兄一起去柏林，在柏林大学学习。他选修了神学、古典语言、哲学和文学课程，但似乎没有选修任何数学或物理课程。

尽管他似乎没有接受过正规的大学数学训练，但正是这门学科在1830年秋他在柏林完成大学学业回到斯德丁后引起了他的兴趣。显然，他父亲的影响对他走上这个方向很重要，此时他决定要成为一名学校教师，但他决心独自进行数学研究。在用一年时间进行数学研究并准备参加中学教学考试后，他于1831年12月前往柏林参加必要的考试。他的论文水平不可能很高，因为考官只给了他一个在中学低年级任教的资格。他被告知，在能够教高年级之前，他需要重新参加考试，并对他所提交的科目表现出更丰富的知识。1832年春，他被任命为斯德丁中学的助理教师。

大约就在这时，他做出了第一批重要的数学发现，这些发现将引导他走向几年后将要发展的重要思想。在其Die Lineale Ausdehnungslehre, ein neuer Zweig der MathematikⓉ（线性扩张论，数学的一个新分支）（1844年）的前言中，格拉斯曼描述了他是如何从1832年左右开始被引向这些思想的。

Die Lineale Ausdehnungslehre, ein neuer Zweig der MathematikⓉ（线性扩张论，数学的一个新分支）前言的一段摘录中，他解释了自己如何做出最初的发现，见THIS LINK。

1834年，格拉斯曼参加了由斯德丁路德宗教会理事会设置的一级神学考试，尽管这可能是他成为路德宗教会牧师的第一步，但他反而在那年秋天前往柏林，就任职业学校（Gewerbeschule）的数学教师。这个空缺之所以出现，是因为前任教师雅各布·施泰纳刚刚被任命为柏林大学的数学讲席。格拉斯曼在职业学校只待了一年，就在他的家乡斯德丁出现了新的机会。一所新学校——奥托学校（Otto Schule）刚刚开办，格拉斯曼被任命教授数学、物理、德语、拉丁语和宗教研究。他只具备低年级任教资格，这在一定程度上解释了他所教科目范围之广。

在接下来的四年里，格拉斯曼非常认真地对待教学工作，但他仍能抽出时间投入数学研究，并集中精力准备进一步的考试。1839年，他通过了由斯德丁路德宗教会理事会设置的二级神学考试，1840年他前往柏林参加考试，以获得在中学最高年级教授某些科目的资格。从那时起，他就能在中学所有年级教授数学、物理、化学和矿物学。

事实上，格拉斯曼在1840年参加的考试在另一方面对他也很重要。作为考试的一部分，他必须提交一篇关于潮汐理论的论文。他从皮埃尔·西蒙·拉普拉斯的Mécanique célesteⓉ（天体力学）和约瑟夫·拉格朗日的Mécanique analytiqueⓉ（分析力学）中采纳了基本理论，但他意识到自己能够应用自1832年以来一直在发展的向量方法（在THIS LINK的Die Lineale Ausdehnungslehre前言中有描述），从而给出一种原创且简化的方法。他的论文Theorie der Ebbe und FlutⓉ（潮汐理论）长达200页，首次引入了基于向量的分析，包括向量加法和减法、向量微分以及向量函数理论。尽管他的论文被考官接受，但他们完全没有看到格拉斯曼所引入创新的重要性。另一方面，这向格拉斯曼表明他的理论具有广泛的适用性，他决定尽可能抽出更多时间来进一步发展他关于向量空间的思想。

当然，格拉斯曼无法将太多时间投入研究，因为他是一位尽职的教师，希望竭尽全力把教学工作做到最好。他写了许多教科书，其中两本于1842年出版：一本关于德语口语，另一本关于拉丁语。写完这些教科书后，他将全部注意力转向撰写Die lineale Ausdehnungslehre, ein neuer Zweig der MathematikⓉ（线性扩张论，数学的一个新分支）。他从1842年春天开始，到1843年秋天完成了手稿。该书于次年出版。在这部必须被视为独创性杰作的著作中，他发展了一种代数的思想，其中代表点、线、面等几何实体的符号按照某些规则进行运算。他用坐标表示空间中的子空间，从而引向一种代数流形的点映射，这种代数现在被称为格拉斯曼流形。

Fearnley-Sander在[27]中写道格拉斯曼在这部作品中提出的向量方法，然后在1862年进一步改进：-

从一组“单位”$e_{1}, e_{2} , e_{3} , ...$开始，他有效地定义了它们生成的自由线性空间；也就是说，他考虑形式线性组合$a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + ...$，其中$a_{j}$是实数，定义加法和实数乘法[以现在通常的方式]，并正式证明这些运算的线性空间性质。……然后他以一种与现代线性代数教材中的表述惊人相似的方式发展了线性无关理论。

他定义了子空间、无关性、张成、维数、子空间的并与交，以及元素到子空间上的投影等概念。他意识到需要证明维数在基变换下的不变性，并做到了这一点。他证明了施泰尼茨交换定理，该定理以1913年发表它的那个人命名……在其他此类结果中，他证明了任何有限集都有一个具有相同张成的无关子集，任何无关集都可扩充为一组基，并且他证明了重要恒等式

dim($U + W$) = dim $U$ + dim$W$ - dim ($U \bigcap W$)。

他得到了基变换下坐标变换的公式，定义了基的初等变换，并证明了每个基变换（等价地，用现代术语说，每个可逆线性变换）都是初等变换的乘积。

格拉斯曼也意识到，一旦几何被置于这种代数形式中，三维空间的表面限制就消失了。格拉斯曼在1844年的AusdehnungslehreⓉ（线性扩张论）中写道：-

如果应用两条不同的变化规则，那么所产生的元素集合……构成一个第二阶段的系统……如果再加入第三条独立的规则，那么就得到一个第三阶段的系统，依此类推。空间理论在这里可以作为一个例子……平面是第二阶段的系统……如果加上第三个独立方向，那么整个无限空间（第三阶段的系统）就被产生出来……在这里不能超过三个独立方向（变化规则），而在纯扩张论中，它们的数量可以增加到无穷。

格拉斯曼发明了现在称为外代数的东西。1878年，威廉·金顿·克利福德将其与威廉·哈密顿的四元数结合起来。威廉·金顿·克利福德用规则取代了格拉斯曼的规则

$e_{p}\land e_{p} = 0$和$e_{p}\land e_{q} = -e_{q}\land e_{p}$用于$p$而非$q$

由规则

$e_{p} e_{p} = 1$和$e_{p} e_{q} = -e_{q} e_{p}$对于$p$不$q$。

威廉·金顿·克利福德代数如今用于二次型理论和相对论性量子力学中。威廉·金顿·克利福德代数与格拉斯曼的外代数一起出现在微分几何中。参见[66]。

数学家们如何看待这部革命性著作？遗憾的是，它远远超前于时代，无法得到赏识。奥古斯特·费迪南德·莫比乌斯不理解格拉斯曼方法的意义，拒绝撰写评论。结果，这本书在很大程度上被忽视了。然而，格拉斯曼继续将他的新概念应用于其他情形，觉得一旦人们看到该理论可以如何应用，就会认真对待它。他于1845年出版了Neue Theorie der ElektrodynamikⓉ（电动力学新理论），并在接下来的十年里写了许多应用于代数曲线和曲面的论文。他因1846年的一部著作而获得最多认可。奥古斯特·费迪南德·莫比乌斯建议他参加Fürstliche Jablonowski'schen Gesellschaft设立的奖项，该奖项面向解决一个问题的参赛作品，这个问题最初由哥特弗里德·威廉·莱布尼茨提出，要求在不使用度量性质的情况下建立几何特征。格拉斯曼提交了Die Geometrische Analyse geknüpft und die von Leibniz CharacteristikⓉ（几何分析及哥特弗里德·威廉·莱布尼茨的特征），并于1846年7月1日获奖。然而，对格拉斯曼来说并非全是好消息，因为他是唯一的参赛者，而评委之一奥古斯特·费迪南德·莫比乌斯批评了格拉斯曼引入抽象概念的方式，认为他没有为读者提供一个可以挂靠这些概念的直观抓手。

格拉斯曼感到有些委屈，因为他正在产出极具创新性的数学，他认为这很重要，但他仍在中学教书。事实上，尽管他自首次被任命到奥托学校以来一直在斯德丁，但由于该镇的教育重组，他先被调到斯德丁文理中学，然后调到弗里德里希·威廉学校。1847年5月，他在弗里德里希·威廉学校获得了高级教师头衔，同月他写信给普鲁士教育部，请求将他列入考虑大学职位的名单。教育部向恩斯特·爱德华·库默尔询问他对格拉斯曼的看法，后者阅读了他的获奖论文Geometrische Analyse Ⓣ（几何分析），并报告说其中包含：-

……值得称赞的良好材料，但表达形式有缺陷。

恩斯特·爱德华·库默尔的报告终结了格拉斯曼可能获得的大学职位的任何希望。有趣的是，有多少领先的数学家未能认识到格拉斯曼所提出的数学将在100年后成为该学科的基本基础。

1848-49年以革命为标志。1848年2月法国国王路易-菲利普被推翻，成为德意志邦联革命的信号。人们采取了走向德国政治统一的行动，但随后就国家应如何治理发生了激烈争论。在1848-49年这段革命时期，格拉斯曼和他的兄弟罗伯特一起出版了一份政治周报。他们的政治立场是推动德国统一为君主立宪制。在撰写了一系列关于宪法的文章后，格拉斯曼与报纸的政治方向越来越不一致，并退出了该报。

1849年早些时候，他于4月12日与地主的女儿特蕾泽·克纳普结婚。他们有十一个孩子，其中七个活到成年。他们的一个儿子，赫尔曼 Ernst Grassmann，于1893年获得博士学位，其学位论文Anwendung der Ausdehnungslehre auf die Allgemeine Theorie der Raumkurven und Krummen Flächen Ⓣ（扩展理论在一般空间曲线和Krummen曲面理论中的应用）是在Albert Wangerin的指导下在哈勒-维滕贝格大学完成的。他后来成为吉森大学的数学教授。

1852年3月，格拉斯曼的父亲尤斯图斯去世，同年晚些时候，格拉斯曼被任命填补他父亲在斯德丁文理中学以前的职位。这意味着，尽管他仍在中学教书，他现在有了教授的头衔。值得注意的是，格拉斯曼的两个儿子，尤斯图斯和马克斯，最终成为斯德丁文理中学的教师。由于他的数学未能获得认可，格拉斯曼转向了他其他最喜欢的科目之一，梵文和哥特语的研究。事实上，在他的一生中，公平地说，他在语言研究方面获得了更多的认可：-

通过证明日耳曼语在某一音系模式上实际上比梵语更“古老”，格拉斯曼削弱了梵语作为印欧语言学中可追溯的最早语言的地位。通过这一证明，格拉斯曼也削弱了语言从分析结构发展为综合结构这一观念，即通过[将简单词组合而不改变其形式来构成新词]。

格拉斯曼还研究过物理学问题，特别是1853年发表了一种颜色混合理论，与赫尔曼·冯·亥姆霍兹提出的理论相矛盾。然而到次年年中，他已回到数学和他的扩展理论，决定不再像原先打算的那样写第二卷，而是彻底重写这部著作，以期使其意义得到承认。事实上，尽管他写的著作在我们今天看来像是现代教科书的风格，格拉斯曼却未能说服同时代的数学家。也许他只是太确信这一主题的重要性，以至于无法说服自己去向怀疑的读者推销它。当然，格拉斯曼于1862年出版的Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet Ⓣ（线性扩展理论：以严格形式完整处理）一书，境遇并不比1844年的第一版更好。

你可以在THIS LINK阅读1862年书籍前言的一部分。

由于无法说服数学家而感到失望，他再次转向语言学研究。在这里他确实做得好得多，他因对这一学术领域的贡献而受到荣誉，被选为美国东方学会会员，并获得图宾根大学的名誉学位。在他生命的最后几年，他确实回到了数学，尽管健康状况不佳，他还是准备了1844年Ausdehnungslehre的另一版本以供出版。它确实出现了，但只在他死后。格拉斯曼在健康缓慢恶化一段时间后死于心脏问题。

格拉斯曼的数学方法被采纳得很慢，但最终启发了埃利·嘉当的工作，此后一直被用于研究微分形式及其在分析和几何中的应用。其他直接受到影响的人包括赫尔曼·汉克尔、朱塞佩·皮亚诺、阿尔弗雷德·诺思·怀特海和菲利克斯·克莱因。正如朱塞佩·皮亚诺本人所承认的，他的许多贡献都基于格拉斯曼的思想。正如A C Lewis所写：-

格拉斯曼的命运似乎就是被不时重新发现，每一次都仿佛他自1879年去世以来几乎已被遗忘。

Fearnley-Sander在[27]中写道：-

所有数学家都如艾萨克·牛顿所说，站在巨人的肩膀上，但很少有人比格拉斯曼更接近于单枪匹马地创造一门新学科。

Hermann Grassmann's father was Justus Günter Grassmann and his mother was Johanne Luise Friederike Medenwald, who was the daughter of a minister from Klein-Schönfeld. Justus had been ordained a minister but he had taken a position in the Gymnasium at Stettin as a teacher of mathematics and physics. He was a fine academic who wrote several school books on physics and mathematics, and also undertook research on crystallography. Johanne and Justus had twelve children, Hermann being their third child. Hermann's brother Robert also became a mathematician and the two collaborated on many projects.

When Hermann was young he was taught by his mother, who was a well educated woman. He then attended a private school before entering the Gymnasium in Stettin where his father taught. Most of the mathematicians in this archive impressed their teachers from a young age, but surprisingly, despite having excellent educational opportunities in an educationally minded family, Hermann did not excel during his first few years at the Gymnasium. His father felt that he should aim at a manual job such as a gardener or a craftsman. Hermann did find pleasure in music and learnt to play the piano. As he progressed through the school he did slowly improve and by the time he took his final secondary school examinations at the age of eighteen, he was ranked second in the school. Having proved himself at least a very competent scholar, Hermann decided that he would study theology, and he went to Berlin in 1827 with his eldest brother to study at the University of Berlin. He took courses on theology, classical languages, philosophy, and literature but does not appear to have taken any courses on mathematics or physics.

Although he seems to have had no formal university training in mathematics, it was this topic which interested him on his return to Stettin in the autumn of 1830 after completing his university studies in Berlin. Clearly his father's influence was important in taking him in that direction, and he decided at this time that he would become a school teacher but he was determined to undertake mathematical research on his own. After a year undertaking research in mathematics and preparing himself to take the examinations to teach in gymnasiums, he went to Berlin in December 1831 to take the necessary examinations. His papers could not have been of a good standard, since his examiners only gave him at a pass to teach at the lower levels of a gymnasium. He was told that before he could teach at higher levels he would need to retake the examinations and show a much greater knowledge of the subjects for which he had presented himself. In the spring of 1832 he was appointed to the Gymnasium at Stettin as an assistant teacher.

It was about this time that he made his first significant mathematical discoveries which were to lead him to the important ideas he would develop a few years later. In the Foreword of his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik Ⓣ (1844) Grassmann described how he was led to these ideas starting around 1832.

An extract from the Foreword of Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik Ⓣ in which he explained how he made his initial discoveries is at THIS LINK.

In 1834 Grassmann took the theology examinations, at level one, set by the Lutheran Church Council of Stettin but although this might have been his first step towards becoming a minister in the Lutheran Church, instead he went to Berlin in the autumn of that year to take up an appointment as a mathematics teacher at the Gewerbeschule. The vacancy had occurred since the previous teacher, Jacob Steiner, had just been appointed to a mathematics chair at the University of Berlin. Grassmann only spent a year at the Gewerbeschule before a new opportunity arose back in his home town of Stettin. A new school, the Otto Schule, had just opened and Grassmann was appointed to teach mathematics, physics, German, Latin, and religious studies. He had only qualified to teach at a low level, and this explains to some extent the wide range of topics he taught.

Over the next four years Grassmann took his teaching very seriously, yet he was able to find time to devote to mathematical research as well as concentrating on preparing himself for further examinations. In 1839 he passed the theology examinations, at level two, set by the Lutheran Church Council of Stettin, and in 1840 he went to Berlin to take examinations which would allow him to teach certain subjects at the highest gymnasium level. From then on he was able to teach mathematics, physics, chemistry and mineralogy at all secondary school levels.

In fact the examinations that Grassmann took in 1840 were significant for him in another way. He had to submit an essay on the theory of the tides as part of the examination. He took the basic theory from Laplace's Mécanique céleste Ⓣ and from Lagrange's Mécanique analytique Ⓣ but he realised that he was able to apply the vector methods which he had been developing since 1832 (described in the preface to Die Lineale Ausdehnungslehre at THIS LINK) to produce an original and simplified approach. His essay Theorie der Ebbe und Flut Ⓣ was 200 pages long and introduced for the first time an analysis based on vectors, including vector addition and subtraction, vector differentiation, and vector function theory. Although his essay was accepted by the examiners they totally failed to see the importance of the innovations which Grassmann had introduced. On the other hand it had shown Grassmann that his theory was widely applicable and he decided to spend as much time as he could spare on further developing his ideas on vector spaces.

Of course Grassmann could not devote too much time to research since he was a dedicated teacher who wanted to put considerable effort into doing that job to the very best of his ability. He wrote a number of textbooks, two of which were published in 1842: one was on spoken German, the other on Latin. After writing these textbooks, he turned his full attention to writing Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik Ⓣ. He started in the spring of 1842 and by the autumn of 1843 he had completed the manuscript. It was published in the following year. In this work, which must be considered as a masterpiece of originality, he developed the idea of an algebra in which the symbols representing geometric entities such as points, lines and planes, are manipulated using certain rules. He represented subspaces of a space by coordinates leading to point mapping of an algebraic manifold now called the Grassmannian.

Fearnley-Sander writes in [27] about the vector methods which Grassmann set out in this work and then refined further in 1862:-

Beginning with a collection of ''units'' $e_{1}, e_{2} , e_{3} , ...$ he effectively defines the free linear space which they generate; that is to say, he considers formal linear combinations $a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3} + ...$ where the $a_{j}$ are real numbers, defines addition and multiplication by real numbers [in what is now the usual way] and formally proves the linear space properties for these operations. ... He then develops the theory of linear independence in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts.

He defines the notions of subspace, independence, span, dimension, join and meet of subspaces, and projections of elements onto subspaces. He is aware of the need to prove invariance of dimension under change of basis, and does so. He proves the Steinitz Exchange Theorem, named for the man who published it in 1913 ... Among other such results, he shows that any finite set has an independent subset with the same span and that any independent set extends to a basis, and he proves the important identity

dim($U + W$) = dim $U$ + dim$W$ - dim ($U \bigcap W$).

He obtains the formula for change of coordinates under change of basis, defines elementary transformations of bases, and shows that every change of basis (equivalently, in modern terms, every invertible linear transformation) is a product of elementaries.

Grassmann also realised that once geometry is put into this algebraic form then the apparent restrictions of 3-dimensional space vanish. Grassmann wrote in the Ausdehnungslehre Ⓣ of 1844:-

If two different rules of change are applied, then the collection of elements produced... forms a system of the second step.... If still a third independent rule is added, then a system of the third step is attained, and so forth. Space theory may serve here as an example.... The plane is the system of the second step.... If one adds a third independent direction, then the whole infinite space (system of the third step) is produced.... One cannot here go further than up to three independent directions (rules of change), while in the pure theory of extension their quantity can increase up to infinity.

Grassmann invented what is now called exterior algebra. This was joined to Hamilton's quaternions by Clifford in 1878. Clifford replaced Grassmann's rules

$e_{p}\land e_{p} = 0$ and $e_{p}\land e_{q} = -e_{q}\land e_{p}$ for $p$ not $q$

by the rules

$e_{p} e_{p} = 1$ and $e_{p} e_{q} = -e_{q} e_{p}$ for $p$ not $q$.

Clifford algebras are used today in the theory of quadratic forms and in relativistic quantum mechanics. Clifford algebras appear together with Grassmann's exterior algebra in differential geometry. See [66].

What did mathematicians make of this revolutionary text? Sadly it was far too much ahead of its time to be appreciated. Möbius did not understand the significance of Grassmann's approach and declined to write a review. As a consequence the book was largely ignored. Grassmann, however, went on to apply his new concepts to other situations, feeling that once people saw how the theory could be applied they would take it seriously. He published Neue Theorie der Elektrodynamik Ⓣ in 1845 and wrote various papers with applications to algebraic curves and surfaces over the next ten years. He received most recognition for a work he produced in 1846. Möbius suggested that he enter for the prize proposed by the Fürstliche Jablonowski'schen Gesellschaft for entries which solved a problem, first proposed by Leibniz, to establish geometric characteristic without using metric properties. Grassmann submitted Die Geometrische Analyse geknüpft und die von Leibniz Characteristik Ⓣ which received the award on 1 July 1846. However, it was not all good news for Grassmann since his was the only entry and Möbius, who was one of the judges, criticised the way that Grassmann introduced abstract ideas without providing the reader with an intuitive hook on which to hang them.

Grassmann felt somewhat aggrieved that he was producing highly innovative mathematics which he felt was important yet he was still teaching in secondary schools. In fact although he had been in Stettin since first appointed to the Otto Schule, he had been moved first to the Stettin Gymnasium, then to the Friedrich Wilhelm Schule due to educational reorganisation in the town. In May 1847 he received the title Oberlehrer at the Friedrich Wilhelm Schule and in the same month he wrote to the Prussian Ministry of Education requesting that he be put on a list of those to be considered for university positions. The Ministry of Education asked Kummer for his opinion of Grassmann who read his prize winning essay Geometrische Analyse Ⓣ and reported that it contained:-

... commendably good material expressed in a deficient form.

Kummer's report ended any hopes that Grassmann might have had to obtain a university post. It is interesting to see just how many leading mathematicians failed to recognise that the mathematics Grassmann presented would become the basic foundation of the subject in 100 years time.

The years 1848-49 were marked by revolutions. The overthrow of King Louis-Philippe of France in February 1848 was the signal for revolutions in the German Confederation. Moves were made towards political unification of Germany but bitter disputes followed as to the way the country should be governed. During this revolutionary period of 1848-49 Grassmann, together with his brother Robert, published a political weekly newspaper. Their political position was one of pressing for the unification of Germany as a constitutional monarchy. After writing a series of articles on constitutional law, Grassmann became increasingly at odds with the political direction the newspaper was going and withdrew form it.

Earlier in 1849 he had married Therese Knappe, the daughter of a landowner, on 12 April. They had eleven children of whom seven reached adulthood. One of their sons, Hermann Ernst Grassmann, received a doctorate in 1893 for his thesis Anwendung der Ausdehnungslehre auf die Allgemeine Theorie der Raumkurven und Krummen Flächen Ⓣ written under Albert Wangerin's supervision at the University of Halle-Wittenberg. He went on to become professor of mathematics at the University of Giessen.

In March 1852 Grassmann's father Justus died and later that year Grassmann was appointed to fill his father's former position at Stettin Gymnasium. This meant that, although still teaching in a secondary school, he now had the title of professor. It is worth noting that two of Grassmann's sons, Justus and Max, eventually became teachers at Stettin Gymnasium. Having failed to gain recognition for his mathematics, Grassmann turned to one of his other favourite subjects, the study of Sanskrit and Gothic. In fact during his life it is fair to say that he gained more recognition for his study of languages:-

By demonstrating that Germanic actually was "older" in one phonological pattern than was Sanskrit, Grassmann undermined the position of Sanskrit as the language which was the earliest attainable in Indo-European linguistics. By this demonstration Grassmann also undermined the notion that language developed from an analytic to a synthetic structure through [combining simple words without changing their form to make new words].

But Grassmann also studied problems in physics, in particular publishing a theory of the mixing of colours in 1853 which contradicted that proposed by Helmholtz. By the middle of the following year, however, he had returned to mathematics and his theory of extension deciding that rather than write a second volume, as he had originally intended, he would completely rewrite the work in an attempt to have its significance recognised. In fact, despite writing a work which appears to us today to be in the style of a modern textbook, Grassmann failed to convince mathematicians of his own time. Perhaps he was just so assured of the importance of the topic that he could not bring himself to set out to sell it to sceptical readers. Certainly the book Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet Ⓣ published by Grassmann in 1862 fared no better that the first version of 1844.

You can read part of the Foreword to the 1862 book at THIS LINK.

Disappointed that he could not convince mathematicians, he turned again to research in linguistics. Here he did indeed fare much better and he was honoured for his contributions to this area of scholarship by being elected to the American Oriental Society, and with the award of an honorary degree by the University of Tübingen. He did return to mathematics in the last couple of years of his life and, despite failing health, prepared another edition of the 1844 Ausdehnungslehre for publication. It did appear, but only after his death. Grassmann died of heart problems after a period of slowly failing health.

Grassmann's mathematical methods were slow to be adopted but eventually they inspired the work of Élie Cartan and have since been used in studying differential forms and their application to analysis and geometry. Other who were directly influenced included Hankel, Peano, Whitehead, and Klein. Much of Peano's contributions were, as he acknowledges himself, based on the ideas of Grassmann. As A C Lewis writes:-

It seems to be Grassmann's fate to be rediscovered from time to time, each time as if he had been virtually forgotten since his death in 1879.

Fearnley-Sander writes in [27]:-

All mathematicians stand, as Newton said he did, on the shoulders of giants, but few have come closer than Hermann Grassmann to creating, single-handedly, a new subject.