赫尔曼·冯·亥姆霍兹Hermann von Helmholtz

赫尔曼·冯·亥姆霍兹的父亲是August Ferdinand Julius Helmholtz，而他的母亲是Caroline Penn。雅各布·赫尔曼是他父母四个孩子中的长子。他的童年对他的性格和后来的职业生涯都有强烈影响。特别是他父亲所持有的哲学观点限制了亥姆霍兹自己的观点。

亥姆霍兹曾在普鲁士军队中服役，对抗拿破仑。尽管他在语文学和哲学方面受过良好的大学教育，但他成为了波茨坦文理中学的一名教师。这份工作薪水微薄，赫尔曼在经济困难的环境中长大。费迪南德是个有艺术气质的人，他的影响意味着赫尔曼从小就热爱音乐和绘画。Caroline Helmholtz是一名炮兵军官的女儿。从她那里，赫尔曼继承了[1]：-

……那种在他晚年性格中显著的平静与含蓄。

赫尔曼就读于波茨坦文理中学，他的父亲在那里教授语文学和古典文学。他在学校的主要兴趣是物理，他本希望在大学学习这门学科。然而，家庭的经济状况意味着他只有获得奖学金才能上大学。这种经济支持只针对特定学科，而赫尔曼的父亲说服他应该学习医学，因为医学受到政府的资助。

1837年，亥姆霍兹获得政府资助，得以在柏林的皇家弗里德里希-威廉医学与外科学院学习医学。然而，这笔钱并非无条件提供，他必须签署一份文件，承诺毕业后在普鲁士军队中担任医生十年。1838年，他开始在柏林学习。尽管他正式就读于医学与外科学院，但身在柏林，他有机会参加大学的课程。他抓住了这个机会，旁听了化学和生理学的讲座。

鉴于亥姆霍兹后来在数学上的贡献，人们本有理由期望他此时在柏林大学选修数学课程。然而他并没有，而是自学数学，阅读皮埃尔·西蒙·拉普拉斯、让-巴蒂斯特·毕奥和丹尼尔·伯努利的著作。他此时还阅读哲学著作，尤其是⟦N4⟧的作品。他的研究生涯始于1841年，当时他开始撰写学位论文。他摒弃了生理学此前所遵循的方向，即基于非物理性质的活力。亥姆霍兹强烈主张将生理学完全建立在物理学和化学原理之上。

亥姆霍兹于1843年从柏林医学研究所毕业，被分配到波茨坦的一个军团，但他把所有业余时间都用于研究。如上所述，他的工作仍集中于证明肌肉力源于化学和物理原理。他论证说，如果存在某种活力，那么永动机就会成为可能。1847年，他在一篇非常重要的论文Über die Erhaltung der KraftⓉ（论能量守恒）中发表了自己的观点，该论文研究了能量守恒背后的数学原理。

亥姆霍兹 既用哲学论证也用物理论证来支持能量守恒。他的许多想法基于 萨迪·卡诺、埃米尔·克拉佩龙、Joule 等人的早期工作。哲学论证在这项工作中被放在最前面，这是 亥姆霍兹 所有贡献的典型特征。他主张物理科学家必须进行实验以发现普遍定律。然后是理论论证（引自该论文）：-

……致力于从过程的可见效果中查明其未知原因；它力求按照因果律来理解它们。……因此，理论自然科学如果不想满足于对事物本性的片面看法，就必须采取一种与当前关于简单力的本性及其后果的概念相协调的立场。当把现象还原为简单力得以完成，并且同时能够证明所给出的还原是现象所允许的唯一可能的还原时，它的任务就完成了。

他表明，功不可能无中生有地不断产生的假设会导致动能守恒。然后他将这一原理应用于各种不同的情形。他证明，在能量似乎损失的各种情形中，它实际上转化成了热能。这发生在碰撞、气体膨胀、肌肉收缩以及其他情形中。该论文考察了大量应用，包括静电学、伽伐尼现象和电动力学。

这篇论文是一项重要贡献，并很快被如此看待。事实上，它在亥姆霍兹的职业生涯中发挥了重要作用，因为第二年他被免除了担任军医的义务，以便接受柯尼斯堡空缺的生理学讲席。他于1849年8月26日与Olga von Velten结婚，并安定下来从事学术生涯。

一方面，他的职业生涯在柯尼斯堡迅速发展。他发表了关于生理光学和生理声学的重要著作。他因1851年发明检眼镜而广受赞誉，并迅速赢得了强大的国际声誉。1852年，他发表了关于生理光学的重要著作，提出了他的色觉理论。然而，他此时进行的实验使他否定了艾萨克·牛顿的颜色理论。这篇论文受到了赫尔曼·格拉斯曼和詹姆斯·克拉克·麦克斯韦的恰当批评。亥姆霍兹总是愿意承认自己的错误，事实上三年后他就这样做了，当时他发表了新的实验结果，表明他1852年论文中的结果是错误的。

1853年访问英国时，他与开尔文建立了重要的友谊。然而，另一方面，柯尼斯堡也存在问题。恩斯特·弗朗茨·诺伊曼，柯尼斯堡的物理学教授，与亥姆霍兹卷入了关于优先权的争执，而柯尼斯堡寒冷的天气对他妻子虚弱的健康产生了不良影响。他请求调动，并于1855年被任命为波恩空缺的解剖学与生理学讲席。

1856年，他出版了他的Handbook of physiological optics第一卷，然后在1858年，他在Crelle's Journal上发表了他关于完美流体运动的重要论文。亥姆霍兹的论文Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechenⓉ（论流体动力学方程的积分，对应于涡旋运动）首先将完美流体的运动分解为平移、旋转和变形。亥姆霍兹将涡线定义为与流体局部旋转轴方向重合的线，并将涡管定义为通过无穷小面积元的涡线束。亥姆霍兹表明涡管必须闭合，并且在任何给定时刻，涡管中的粒子将无限期地留在管内，因此无论管子被扭曲多少，它都会保持其形状。

亥姆霍兹 在他的论文中知道 topological 的想法，特别是涡管外部的区域是多连通的这一事实，这使他考虑多值势函数。他以下列方式描述了他关于两个具有共同对称轴的圆形涡环的理论结论：-

如果它们具有相同的旋转方向，它们将沿同一方向前进，前面的环会变大并移动得更慢，而第二个环会缩小并移动得更快；如果平动速度相差不太大，第二个环最终会追上第一个环并穿过它。然后同样的过程会与另一个环重复，因此环会交替地相互穿过。

这篇论文在数学方法上极为严谨，当时并未引起太多注意，但它对 彼得·格思里·泰特 和 开尔文 未来工作的影响非常显著。关于这项工作影响的细节，特别是 亥姆霍兹 关于涡旋的结果，见文章 拓扑学与苏格兰数学物理。

在这篇论文发表之前，亥姆霍兹已经对他在波恩的新职位感到不满。问题的一部分似乎围绕着这样一个事实：该讲席涉及解剖学，并且有人向教育部长投诉他关于这一主题的讲座不称职。亥姆霍兹对这些批评反应强烈，他认为这些批评是由传统主义者提出的，他们不理解他对该主题的新力学方法。对于亥姆霍兹来说，这是一个有些奇怪的处境，因为他作为世界领先科学家的声誉非常强大。当1857年海德堡的讲席提供给他时，他并没有立即接受。当1858年提出进一步的甜头来诱使他接受时，例如承诺建立一个新的生理学研究所，亥姆霍兹同意了。

亥姆霍兹遭遇了一些个人问题。他的父亲于1858年去世，接着在1859年底，他健康状况一直不佳的妻子也去世了。他不得不独自抚养两个年幼的孩子，并在十八个月内再次结婚。1861年5月16日，亥姆霍兹与安娜·冯·莫尔结婚，她是海德堡另一位教授[1]的女儿：-

安娜后来为亥姆霍兹生了三个孩子，她是一位迷人、世故的女性，比丈夫年轻得多。这段婚姻为亥姆霍兹开启了一段更广泛社交接触的时期。

他在海德堡担任此职位期间完成了一些最重要的工作。他研究数学物理和声学，在1862年发表了一项重要研究，探讨了音乐理论和声音的感知。在数学附录中，他提倡使用Fourier series。1843年，格奥尔格·欧姆提出了生理声学的基本原理，涉及人如何听到组合音的方式。亥姆霍兹基于他的基本生理假设解释了音乐的起源。他提出了听觉的共振理论，为格奥尔格·欧姆的原理提供了生理学解释。他对音乐理论的贡献在[8]中有充分讨论。

大约从1866年开始，亥姆霍兹开始从生理学转向更多地向物理学发展。当1870年柏林物理学讲席空缺时，他表示了对该职位的兴趣。古斯塔夫·基尔霍夫是另一位主要候选人，由于他被认为是一位比亥姆霍兹更优秀的教师，因此获得了该职位。然而，当古斯塔夫·基尔霍夫决定不接受时，亥姆霍兹处于有利地位。他能够谈判获得高薪，并让普鲁士同意在柏林建立一个新的物理研究所，由亥姆霍兹控制。1871年，他担任了这一职位。

亥姆霍兹大约在1867年他的兴趣转向物理学时开始研究非欧几里得空间的性质。贝尔纳多在[9]中写道：-

在19世纪下半叶，科学家和哲学家们卷入了一场关于几何学原理以及所谓非欧几何有效性的激烈讨论。……亥姆霍兹关于这一主题的研究始于1867年至1868年间。从观察到我们的几何能力依赖于自然界中刚体的存在出发，他推测自己已经证明了欧几里得几何是唯一与这些刚体相容的几何，同时坚持几何学的经验起源而非先验起源。1869年，在收到贝尔特拉米的信之后……他意识到自己犯了一个错误：刚体的经验概念和数学本身不足以刻画欧几里得几何。第二年，他完全认同了通过卡尔·弗里德里希·高斯、波恩哈德·黎曼、罗巴切夫斯基和贝尔特拉米导致新几何学创立的数学进程，他提议在哲学家中间传播这一知识，同时批评康德体系。这标志着一场激烈的哲学讨论的开始，这场讨论导致亥姆霍兹在1878年试图平息对康德先验论的批评。

亥姆霍兹被任命到柏林后关注的一个主要主题是电动力学。他与威廉·韦伯讨论了威廉·韦伯的电动力学与能量守恒原理的相容性。事实上，这场争论很激烈，持续了整个1870年代。这是一场双方都没有真正获胜的争论，到了1880年代，詹姆斯·克拉克·麦克斯韦的理论被接受了。亥姆霍兹试图为热力学提供力学基础，他还试图从最小作用量原理推导出詹姆斯·克拉克·麦克斯韦的电磁场方程。

R Steven Turner 在1中写道：-

亥姆霍兹毕生致力于探寻自然界背后那些伟大的统一原理。他的学术生涯始于这样一个原理，即能量原理，而终于另一个原理，即最小作用量原理。与他之前那一代理想主义者一样，他渴望理解知识的终极的、主观的来源。这种渴望体现在他决心去理解感觉器官作为经验的中介在知识综合中所起的作用。

对于与过去的这种连续性，亥姆霍兹和他那一代人带来了两个新元素：对形而上学的深刻厌恶，以及对数学和力学机制的坚定不移的依赖。亥姆霍兹最伟大著作所特有的广度和深度，在很大程度上归功于他带入科学中的数学和实验专长。……亥姆霍兹是最后一位伟大的学者，其工作遵循哥特弗里德·威廉·莱布尼茨的传统，涵盖了所有科学以及哲学和美术。

Hermann von Helmholtz's father was August Ferdinand Julius Helmholtz while his mother was Caroline Penn. Hermann was the eldest of his parents four children. His childhood had a strong influence on both his character and his later career. In particular the views on philosophy held by his father restricted Helmholtz's own views.

Ferdinand Helmholtz had served in the Prussian army in the fight against Napoleon. Despite having a good university education in philology and philosophy, he became a teacher at Potsdam Gymnasium. It was a poorly paid job and Hermann was brought up in financially difficult circumstances. Ferdinand was an artistic man and his influence meant that Hermann grew up to have a strong love of music and painting. Caroline Helmholtz was the daughter of an artillery officer. From her Hermann inherited [1]:-

... the placidity and reserve which marked his character in later life.

Hermann attended Potsdam Gymnasium where his father taught philology and classical literature. His interests at school were mainly in physics and he would have liked to have studied that subject at university. The financial position of the family, however, meant that he could only study at university if he received a scholarship. Such financial support was only available for particular topics and Hermann's father persuaded him that he should study medicine which was supported by the government.

In 1837 Helmholtz was awarded a government grant to enable him to study medicine at the Royal Friedrich-Wilhelm Institute of Medicine and Surgery in Berlin. He did not receive the money without strings attached, however, and he had to sign a document promising to work for ten years as a doctor in the Prussian army after graduating. In 1838 he began his studies in Berlin. Although he was officially studying at the Institute of Medicine and Surgery, being in Berlin he had the opportunity of attending courses at the University. He took this chance, attending lectures in chemistry and physiology.

Given Helmholtz's contributions to mathematics later in his career it would be reasonable to have expected him to have taken mathematics courses at the University of Berlin at this time. However he did not, rather he studied mathematics on his own, reading works by Laplace, Biot and Daniel Bernoulli. He also read philosophy works at this time, particularly the works of Kant. His research career began in 1841 when he began work on his dissertation. He rejected the direction which physiology had been taking which had been based on vital forces which were not physical in nature. Helmholtz strongly argued for founding physiology completely on the principles of physics and chemistry.

Helmholtz graduated from the Medical Institute in Berlin in 1843 and was assigned to a military regiment at Potsdam, but spent all his spare time doing research. His work still concentrated, as we remarked above, on showing that muscle force was derived from chemical and physical principles. If some vital force were present, he argued, then perpetual motion would become possible. In 1847 he published his ideas in a very important paper Über die Erhaltung der Kraft Ⓣ which studied the mathematical principles behind the conservation of energy.

Helmholtz argued in favour of the conservation of energy using both philosophical arguments and physical arguments. He based many ideas on the earlier works by Sadi Carnot, Clapeyron, Joule and others. That philosophical arguments came right up front in this work was typical of all of Helmholtz's contributions. He argued that physical scientists had to conduct experiments to find general laws. Then theoretical argument (quoting from the paper):-

... endeavours to ascertain the unknown causes of processes from their visible effects; it seeks to comprehend them according to the laws of causality. ... Theoretical natural science must, therefore, if it is not to rest content with a partial view of the nature of things, take a position in harmony with the present conception of the nature of simple forces and the consequences of this conception. Its task will be completed when the reduction of phenomena to simple forces is completed, and when it can at the same time be proved that the reduction given is the only one possible which the phenomena will permit.

He showed that the assumption that work could not continually be produced from nothing led to the conservation of kinetic energy. This principle he then applied to a variety of different situations. He demonstrated that in various situations where energy appears to be lost, it is in fact converted into heat energy. This happens in collisions, expanding gases, muscle contraction, and other situations. The paper looks at a broad number of applications including electrostatics, galvanic phenomena and electrodynamics.

The paper is an important contribution and it was quickly seen as such. In fact it played a large role in Helmholtz's career for the following year he was released from his obligation to serve as an army doctor so that he could accept the vacant chair of physiology at Königsberg. He married Olga von Velten on 26 August 1849 and settled down to an academic career.

On one hand his career progressed rapidly in Königsberg. He published important work on physiological optics and physiological acoustics. He received great acclaim for his invention of the ophthalmoscope in 1851 and rapidly gained a strong international reputation. In 1852 he published important work on physiological optics with his theory of colour vision. However, experiments which he carried out at this time led him to reject Newton's theory of colour. The paper was rightly criticised by Grassmann and Maxwell. Helmholtz was always prepared to admit his mistakes and indeed he did just this three years later when he published new experimental results showing those of his 1852 paper to be incorrect.

A visit to Britain in 1853 saw him form an important friendship with William Thomson. However, on the other hand, there were problems in Königsberg. Franz Neumann, the professor of physics in Königsberg was involved in disputes concerning priority with Helmholtz and the cold weather in Königsberg had a bad effect on his wife's delicate health. He requested a move and, in 1855, was appointed to the vacant chair of anatomy and physiology in Bonn.

In 1856 he published the first volume of his Handbook of physiological optics, then in 1858 he published his important paper in Crelle's Journal on the motion of a perfect fluid. Helmholtz's paper Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen Ⓣ began by decomposing the motion of a perfect fluid into translation, rotation and deformation. Helmholtz defined vortex lines as lines coinciding with the local direction of the axis of rotation of the fluid, and vortex tubes as bundles of vortex lines through an infinitesimal element of area. Helmholtz showed that the vortex tubes had to close up and also that the particles in a vortex tube at any given instant would remain in the tube indefinitely so no matter how much the tube was distorted it would retain its shape.

Helmholtz was aware of the topological ideas in his paper, particularly the fact that the region outside a vortex tube was multiply connected which led him to consider many-valued potential functions. He described his theoretical conclusions regarding two circular vortex rings with a common axis of symmetry in the following way:-

If they both have the same direction of rotation they will proceed in the same sense, and the ring in front will enlarge itself and move slower, while the second one will shrink and move faster, if the velocities of translation are not too different, the second will finally reach the first and pass through it. Then the same game will be repeated with the other ring, so the ring will pass alternately one through the other.

This paper, highly rigorous in its mathematical approach, did not attract much attention at the time but its impact on the future work by Tait and Thomson was very marked. For details of the impact of this work, particularly Helmholtz's results on vortices, see the article Topology and Scottish mathematical physics.

Before the publication of this paper Helmholtz had become unhappy with his new position in Bonn. Part of the problem seemed to revolve round the fact that the chair involved anatomy and complaints were made to the Minister of Education that his lectures on this topic were incompetent. Helmholtz reacted strongly to these criticisms which, he felt, were made by traditionalists who did not understand his new mechanical approach to the subject. It was a somewhat strange position for Helmholtz to be in for he had a very strong reputation as a leading world scientist. When he was offered the chair in Heidelberg in 1857, he did not accept at once however. When further sweeteners were put forward in 1858 to entice him to accept, such as the promise of setting up a new Physiology Institute, Helmholtz agreed.

Helmholtz suffered some personal problems. His father died in 1858, then at the end of 1859 his wife, whose health had never been good, died. He was left to bring up two young children and within eighteen months he married again. On 16 May 1861 Helmholtz married Anna von Mohl, the daughter of another professor at Heidelberg [1]:-

Anna, by whom Helmholtz later had three children, was an attractive, sophisticated woman considerably younger than her husband. The marriage opened a period of broader social contacts for Helmholtz.

Some of his most important work was carried out while he held this post in Heidelberg. He studied mathematical physics and acoustics producing a major study in 1862 which looked at musical theory and the perception of sound. In mathematical appendices he advocated the use of Fourier series. In 1843 Ohm had stated the fundamental principle of physiological acoustics, concerned with the way in which one hears combination tones. Helmholtz explained the origin of music on the basis of his fundamental physiological hypotheses. He formulated a resonance theory of hearing which provided a physiological explanation of Ohm's principle. His contributions to the theory of music are discussed fully in [8].

From around 1866 Helmholtz began to move away from physiology and move more towards physics. When the chair of physics in Berlin became vacant in 1870 he indicated his interest in the position. Kirchhoff was the other main candidate and because he was considered a superior teacher to Helmholtz he was offered the post. However, when Kirchhoff decided not to accept Helmholtz was in a strong position. He was able to negotiate a high salary as well as having Prussia agree to build a new physics institute under Helmholtz control in Berlin. In 1871 he took up this post.

Helmholtz had begun to investigate the properties of non-Euclidean space around the time his interests were turning towards physics in 1867. Bernardo in [9] writes:-

In the second half of the 19th century, scientists and philosophers were involved in a heated discussion on the principles of geometry and on the validity of so-called non-Euclidean geometry. ... Helmholtz's research on the subject began between 1867 and 1868. Moving from the observation that our geometric faculties depend on the existence, in nature, of rigid bodies, he presumed he had given a proof that Euclidean geometry was the only one compatible with these bodies, maintaining, at the same time, the empirical, not a priori, origin of geometry. In 1869, after Beltrami's letter ... he realized he had made a mistake: the empirical concept of a rigid body and mathematics alone were not enough to characterize Euclidean geometry. The following year, fully sharing the mathematical itinerary that, through Gauss, Riemann, Lobachevsky and Beltrami, led to the creation of the new geometry, he proposed to spread this knowledge among philosophers while at the same time criticizing the Kantian system. This marked the beginning of a heated philosophical discussion that led Helmholtz in 1878 to try to appease the criticisms of the Kantian a priori.

A major topic which occupied Helmholtz after his appointment to Berlin was electrodynamics. He discussed with Weber the compatibility of Weber's electrodynamics with the principle of the conservation of energy. In fact the argument was heated and lasted throughout the 1870s. It was an argument which neither really won and the 1880s saw Maxwell's theory accepted. Helmholtz attempted to give a mechanical foundation to thermodynamics, and he also tried to derive Maxwell's electromagnetic field equations from the least action principle.

R Steven Turner writes in [1]:-

Helmholtz devoted his life to seeking the great unifying principles underlying nature. His career began with one such principle, that of energy, and concluded with another, that of least action. No less than the idealistic generation before him, he longed to understand the ultimate, subjective sources of knowledge. That longing found expression in his determination to understand the role of the sense organs, as mediators of experience, in the synthesis of knowledge.

To this continuity with the past Helmholtz and his generation brought two new elements, a profound distaste for metaphysics and an undeviating reliance on mathematics and mechanism. Helmholtz owed the scope and depth characteristic of his greatest work largely to the mathematical and experimental expertise which he brought to science. ... Helmholtz was the last great scholar whose work, in the tradition of Leibniz, embraced all the sciences, as well as philosophy and the fine arts.