弗里德里希·威廉·贝塞尔Friedrich Bessel

弗里德里希·威廉·贝塞尔的父亲是明登的一名公务员。贝塞尔在明登的文理中学学习了四年，但他似乎并不十分有天赋，觉得拉丁语很难。他后来精通拉丁语，自学了这门语言，这一事实可能表明这所文理中学未能激发贝塞尔。1799年1月，14岁的他离开学校，到不来梅的库伦坎普商行当学徒。这家商行从事进出口业务。

起初贝塞尔从商行没有薪水，但随着他的会计技能得到商行的赏识，他获得了少量薪水。对商行所涉国家的兴趣促使贝塞尔利用晚上学习地理、西班牙语和英语。他的兴趣转向航海，并思考了在海上确定船只位置的问题。这又促使他学习天文学和数学，并开始进行观测以确定经度。

1804年，贝塞尔写了一篇关于爱德蒙·哈雷彗星的论文，利用托马斯·哈里奥特在1607年所作观测的数据计算轨道。他把结果寄给了当时首屈一指的彗星专家海因里希·奥伯斯，奥伯斯立刻认出了弗里德里希·威廉·贝塞尔工作的质量，并给贝塞尔布置了进一步观测的任务，以推进他的工作。由此产生的论文达到了博士学位论文所要求的水平，经奥伯斯推荐发表。从那时起，贝塞尔专注于天文学、天体力学和数学。

奥伯斯向当时仍是那家进出口商行学徒的贝塞尔建议，他应该成为一名职业天文学家。1806年，他接受了不来梅附近利林塔尔天文台助理的职位，这是一家私人天文台。贝塞尔经过相当一番考虑后，才放弃了商业工作中 guaranteed 的富裕生活，转而选择了天文台职位近乎贫困的待遇。然而，利林塔尔天文台给了他观测行星的宝贵经验，特别是土星及其光环和卫星。他还观测彗星，并继续研究天体力学。1807年，他开始着手归算詹姆斯·布拉德利（詹姆斯·布拉德利于1742年至1762年任英国皇家天文学家）约1750年在格林尼治对3222颗恒星位置所作的观测。

贝塞尔的杰出工作很快得到认可，莱比锡和格赖夫斯瓦尔德都向他提供了职位。然而他两处都谢绝了。1809年，26岁的贝塞尔被任命为普鲁士国王腓特烈·威廉三世新建的柯尼斯堡天文台台长和天文学教授。贝塞尔若未先获得博士学位，就不可能获得教授职位。哥廷根大学根据卡尔·弗里德里希·高斯的推荐授予了博士学位，卡尔·弗里德里希·高斯于1807年在不来梅结识了贝塞尔，并认出了他的才华。

尽管柯尼斯堡天文台仍在建设中，贝塞尔于1810年5月10日就任新职。从1810年到1813年，天文台的建设继续进行，他则继续研究詹姆斯·布拉德利的观测资料。贝塞尔的工作此时已闻名国际，他因基于詹姆斯·布拉德利的观测编制的折射表而荣获法兰西研究院颁发的拉朗德奖。同样在此期间，1812年，他当选为柏林科学院。

柯尼斯堡天文台于1813年建成，贝塞尔开始在那里进行观测。Fricke在1中写道：-

贝塞尔余生都留在柯尼斯堡，不间断地进行研究和教学，尽管他常常抱怨由于气候不利，观测的可能性有限。他拒绝了柏林天文台台长的职位，因为担心会承担更大的行政和社会责任……

正是在柯尼斯堡，贝塞尔承担了测定超过50000颗恒星的位置和自行这一艰巨任务，这导致了1838年天鹅座61的视差的发现。然而，尽管他在1812年缔结了幸福的婚姻（[1]），他的生活并不十分顺遂：-

……他们有两个儿子和三个女儿。[这段婚姻]因疾病和两个儿子的早逝而蒙上阴影。……从1840年起，贝塞尔的健康状况恶化。他最后一次长途旅行是在1842年前往英格兰，在那里他参加了在曼彻斯特举行的英国协会大会。他与包括赫歇尔在内的重要英国科学家的会面给他留下了深刻印象，并促使他尽管健康衰弱，仍完成并出版了一系列著作。两年后，他死于癌症……

让我们更详细地考察贝塞尔的工作。他利用詹姆斯·布拉德利的数据给出了恒星和行星位置的参考系统，也用于确定恒星的位置。他必须推断出詹姆斯·布拉德利仪器的误差以及由折射引起的误差。他必须将位置归算到一个固定日期，并消除地球运动、地球岁差及其他效应的影响。[5]的作者们写道：-

贝塞尔是最早意识到以下一点的天文学家之一：在完全信赖一次位置观测之前，必须对可能进入最终结果的每一种误差都有定量的了解。他之所以使用詹姆斯·布拉德利和内维尔·马斯基林十八世纪在格林尼治的观测，是因为这两位天文学家最早对他们自身的仪器误差，以及进行测量时所处大气的温度和气压，提供了详尽的分析。通过消除所有误差来源——光学的、机械的和气象的——贝塞尔能够从天文结果中获得惊人的精确度，并从中提取出大量新数据。

贝塞尔在确定岁差、章动和光行差常数方面的工作为他赢得了更多荣誉，例如1815年来自柏林科学院的奖项。贝塞尔于1818年在一部给出恒星自行的著作中发表了詹姆斯·布拉德利的恒星位置。1825年，他荣膺皇家学会会士。

1830年，贝塞尔发表了38颗恒星在1750—1850年这100年间的平均位置和视位置。这38颗恒星是内维尔·马斯基林的36颗“基本星”再加上两颗更靠近极区的星。从内维尔·马斯基林的36颗基本星中的天狼星和南河三的自行周期性变化中，贝塞尔推断它们有未被观测到的伴星在轨道上运行。1841年他宣布天狼星有一颗伴星，从而成为第一个预言“暗星”存在的人。十年后，伴星的轨道被计算出来，并于1862年被观测到。

贝塞尔利用视差确定到天鹅座61的距离，并于1838年公布了他的结果。显然，要成功就必须选择一颗离太阳近的恒星。他选择恒星的方法基于他自己的数据，因为他选择了所研究过的所有恒星中自行最大的那颗，并正确地推断这意味着该恒星距离较近。由于天鹅座61是一颗相对暗淡的恒星，这是一个大胆的选择，基于他对自行原因的正确理解。贝塞尔使用夫琅禾费量日仪进行测量，公布了他的数值0.314″，结合地球轨道的直径，得出距离约为10光年。天鹅座61视差的正确值为0.292″。

约翰·赫歇尔得知贝塞尔的成就后，写信给他，将其描述为：-

……实用天文学所见证过的最伟大、最辉煌的胜利。

奥尔伯斯在贝塞尔80岁生日时得知他的成就，说这是一份礼物：-

……首次将我们关于宇宙的观念置于可靠的基础上。

Royal Astronomical Society授予他金质奖章以表彰这一成就。

贝塞尔还研究出一种数学分析方法，涉及现在所谓的Bessel function。他在1817年研究约翰内斯·开普勒的一个问题时引入了这一方法，该问题是确定三个在相互引力作用下运动的物体的运动。这一数学成就在[1]中描述如下：-

贝塞尔 也是一位杰出的数学家，他的名字通过一类特殊的函数而广为人知，这些函数已成为应用数学、物理学和工程学中不可或缺的工具。对这些函数的兴趣……源于处理行星系统中的摄动问题。

贝塞尔 函数作为行星间接摄动的级数展开中的系数出现，即由摄动天体引起的太阳运动所导致的运动。1824年，他在研究行星摄动时更充分地发展了 贝塞尔 函数，并在柏林出版了一部关于它们的论著。这些函数的特殊情形并非首次出现，雅各布·伯努利、丹尼尔·伯努利、莱昂哈德·欧拉 和 约瑟夫·拉格朗日 此前已研究过它们的特殊情形。事实上，很可能正是 约瑟夫·拉格朗日 关于椭圆轨道的工作首次促使 贝塞尔 研究 贝塞尔 函数。

这位14岁便离开正规教育的非凡人物，其贡献超越了天文学和数学。他对大地测量学的贡献包括[2]：-

……1826年对秒摆的一项修正，其长度经过精确计算，使得摆动一次恰好需要一秒。在1831-32年间，他指导了东普鲁士子午线弧的大地测量，并在1841年推导出地球椭率的一个值 $\large\frac{1}{299}\normalsize$，即地球形状偏离完美球体的椭圆畸变程度。

贝塞尔 尽管从未接受过大学教育，却对大学教学产生了非常显著的影响。在[8]中，菲利克斯·克莱因 描述了 贝塞尔 的名字，连同 卡尔·古斯塔夫·雅各布·雅可比 和 恩斯特·弗朗茨·诺伊曼 的名字，如何与大学教学改革紧密相连，这场改革最初在德国，随后遍及全世界。

Wilhelm Bessel's father was a civil servant in Minden. Bessel attended the Gymnasium in Minden for four years but he did not appear to be very talented, finding Latin difficult. The fact that he later became proficient in Latin, teaching himself the language, probably suggests that the Gymnasium failed to inspire Bessel. In January 1799, at the age of 14, he left school to become an apprentice to the commercial firm of Kulenkamp in Bremen. The firm was involved in the import-export business.

At first Bessel received no salary from the firm but, as his accounting skills became appreciated by the firm, he received a small salary. Interest in the countries his firm dealt with led Bessel to spend his evenings studying geography, Spanish and English. His interests turned towards navigation and he considered the problem of finding the position of a ship at sea. This in turn led him to study astronomy and mathematics and he began to make observations to determine longitude.

In 1804 Bessel wrote a paper on Halley's comet, calculating the orbit using data from observations made by Harriot in 1607. He sent his results to Heinrich Olbers, the leading comet expert of his time, who recognised at once the quality of Bessel's work and Olbers gave Bessel the task of making further observations to carry his work further. The resulting paper, at the level required for a doctoral dissertation, was published on Olbers' recommendation. From that time on Bessel concentrated on astronomy, celestial mechanics and mathematics.

Olbers suggested to Bessel, who was still an apprentice to the import-export firm, that he should become a professional astronomer. In 1806 he accepted the post of assistant at the Lilienthal Observatory, a private observatory near Bremen. It was only after some considerable thought that Bessel left the affluence that was guaranteed in his commercial job choosing instead the near poverty of the Observatory post. However the Lilienthal Observatory gave him valuable experience observing planets, in particular Saturn, its rings and satellites. He also observed comets and continued his study of celestial mechanics. In 1807 he began to work on reducing James Bradley's observations (Bradley was English Astronomer Royal from 1742 to 1762) of the positions of 3222 stars made around 1750 at Greenwich.

Bessel's brilliant work was quickly recognised and both Leipzig and Greifswald offered him posts. However he declined both. In 1809, at the age of 26, Bessel was appointed director of Frederick William III of Prussia's new Königsberg Observatory and professor of astronomy. It was not possible for Bessel to receive a professorship without first being granted the title of doctor. A doctorate was awarded by the University of Göttingen on the recommendation of Gauss, who had met Bessel in Bremen in 1807 and recognised his talents.

Although the Observatory at Königsberg was still under construction, Bessel took up his new post on 10 May 1810. He continued to work on Bradley's observations while work continued on the observatory from 1810 to 1813. Bessel's work had now become known internationally and he was honoured with the award of the Lalande Prize from the Institut de France for his tables of refraction based on Bradley's observations. Also during this period, in 1812, he was elected to the Berlin Academy.

The Königsberg Observatory was completed in 1813 and Bessel began observing there. Fricke writes in [1]:-

Bessel remained in Königsberg for the rest of his life, pursuing his research and teaching without interruption, although he often complained about the limited possibilities for observations because of the unfavourable climate. He declined the directorship of the Berlin Observatory, fearing greater administrative and social responsibilities...

It was in Königsberg that Bessel undertook his monumental task of determining the positions and proper motions of over 50000 stars which led to the discovery in 1838 of the parallax of 61 Cygni. However his life did not run very smoothly although he made a happy marriage in 1812 ( [1]):-

... they had two sons and three daughters. [The marriage] was clouded by sickness and by the early death of both sons. ... From 1840 on, Bessel's health deteriorated. His last long trip, in 1842, was to England, where he participated in the Congress of the British Association in Manchester. His meeting with important English scientists, including Herschel, impressed him deeply and stimulated him to finish and publish, despite his weakened health, a series of works. After two years he died of cancer...

Let us examine Bessel's work in more detail. He used Bradley's data to give a reference system for the positions of stars and planets and also to determine the positions of stars. He had to deduce errors in Bradley's instruments and errors caused by refraction. He had to reduce the positions to one fixed date and eliminate the effects of the Earth's motion, the precession of the Earth and other effects. The authors of [5] write:-

Bessel was one of the first astronomers to realise that, before a positional observation could be fully relied upon, one must have quantitative knowledge of every possible error that might enter into the finished result. He came to use Bradley's and Maskelyne's eighteenth-century Greenwich observations because these two astronomers were the first to provide exhaustive analyses of their own instrumental errors, along with temperature and pressure of the atmosphere through which the measurements had been made. By eliminating all sources of error - optical, mechanical and meteorological - Bessel was able to obtain astronomical results of astonishing delicacy from which a great deal of new data could be extracted.

Bessel's work in determining the constants of precession, nutation and aberration won him further honours, such as a prize from the Berlin Academy in 1815. Bessel published Bradley's stellar positions in 1818 in a work which gives the proper motion of stars. In 1825 he was honoured by election as a Fellow of the Royal Society.

In 1830 Bessel published the mean and apparent positions of 38 stars over the 100 year period 1750-1850. These 38 stars were the 36 'fundamental stars' of Maskelyne together with two further polar stars. From periodic variations in the proper motions of Sirius and Procyon, two of Maskelyne's 36 fundamental stars, Bessel deduced that they had companion stars in orbit which had not been observed. He announced that Sirius had a companion in 1841 thus being the first to predict the existence of 'dark stars'. Ten years later the orbit of the companion was computed and it was observed in 1862.

Bessel used parallax to determine the distance to 61 Cygni announcing his result in 1838. Clearly to succeed it was important to choose a star which was close to the Sun. His method of selecting a star was based on his own data for he chose the star which had the greatest proper motion of all the stars he had studied, correctly deducing that this would mean that the star was nearby. Since 61 Cygni is a relatively dim star it was a bold choice based on his correct understanding of the cause of the proper motions. Bessel, using a Fraunhofer heliometer to make the measurements, announced his value of 0.314" which given the diameter of the Earth's orbit, gave a distance of about 10 light years. The correct value of the parallax of 61 Cygni is 0.292".

John Herschel, when he learnt of Bessel's achievement, wrote to him describing it as:-

... the greatest and most glorious triumph which practical astronomy has ever witnessed.

Olbers, told of Bessel's achievement on his 80th birthday, said it was a gift that:-

... put our ideas about the universe for the first time on a sound basis.

The Royal Astronomical Society awarded him their gold medal to mark this achievement.

Bessel also worked out a method of mathematical analysis involving what is now known as the Bessel function. He introduced this in 1817 in his study of a problem of Kepler of determining the motion of three bodies moving under mutual gravitation. This mathematical achievement is described in [1] as follows:-

Bessel was also an outstanding mathematician whose name became generally known through a special class of functions that have become an indispensable tool in applied mathematics, physics and engineering. The interest in these functions ... arose in the treatment of the problem of the perturbation in the planetary system.

Bessel functions appear as coefficients in the series expansion of the indirect perturbation of a planet, that is the motion caused by the motion of the Sun caused by the perturbing body. In 1824 he developed Bessel functions more fully in a study of planetary perturbations and published a treatise on them in Berlin. It was not the first time that special cases of the functions had appeared, Jacob Bernoulli, Daniel Bernoulli, Euler and Lagrange having studied special cases of them earlier. In fact it was probably Lagrange's work on elliptical orbits that first suggested to Bessel to work on the Bessel functions.

This remarkable man who left formal education at the age of 14 made contributions beyond astronomy and mathematics. His contributions to geodesy include [2]:-

... a correction in 1826 to the seconds pendulum, the length of which is precisely calculated so that it requires exactly one second for a swing. During 1831-32 he directed geodetical measurements of meridian arcs in East Prussia, and in 1841 he deduced a value of $\large\frac{1}{299}\normalsize$ for the ellipticity of the Earth, the amount of elliptical distortion by which the Earth's shape departs from a perfect sphere.

Bessel also had a very significant impact on university teaching despite the fact that he never had a university education. In [8] Klein describes how the name of Bessel, together with the names of Jacobi and Franz Neumann, is intimately linked to the reform of teaching at universities, first in Germany and then throughout the world.