托比亚斯·梅耶Tobias Mayer

托比亚斯·梅耶是一位自学成才的数学家。虽然他出生在马尔巴赫，但他在埃斯林根长大，生活条件非常贫困。他第一部出版的作品是Neue und allgemeine Art, alle Aufgaben aus der Geometrie vermittelst der geometrischen Linien leichter insbesondere wie alle reguläre und irreguläre Vielecke, davon ein Verhältnis ihrer Seiten gegeben, in den Circul geometrisch sollen eingeschrieben werden, sammt einer kurzen nötigen Buchstaben-Rechnenkunst und GeometrieⓉ（以新的和一般的方式，所有几何任务通过几何线轻松计算；特别是对于圆内包含的所有正多边形和不规则多边形，以几何方式，其中有其边之比，连同关于计算艺术和几何的简要必要信件），该书于1741年在埃斯林根出版。让我们注意，这本书出版时他只有十八岁，这对任何人来说都是一项非凡的成就，尤其是对来自贫困家庭的人来说。Eric Forbes在[8]中写道：-

梅耶对数学的兴趣似乎源于他阅读的两本书：Christian von Wolff的《Anfangs-Gründe aller mathematuscher Wissenschaften》Ⓣ（《所有数学科学的初步基础》）和雅克·夏尔·弗朗索瓦·施图姆的《Mathesis enucleata》Ⓣ（《数学阐明》）。无论如何，在他十八岁生日之际，他为自己第一部印刷作品——一篇关于代数方法应用于初等和高等几何问题的论文——撰写前言时，承认了这两本书对他的影响。将梅耶的书与其他书进行比较表明，他的数学技巧知识依赖于Wolff，而他的表述模式则依赖于Sturm。前者给出的确定不规则多边形（或不规则形状田地）面积的方法，被梅耶在其下一部重要著作《Mathematischer Atlas》Ⓣ（《数学地图集》）（奥格斯堡，1745年）中加以说明。...

他从1746年起在纽伦堡受雇为制图师，当时他开始为Homann Heirs公司工作，该公司继承了著名制图师Johann Baptist Homann在纽伦堡建立的公司。1749年，在为Homann公司工作期间，梅耶制作了一幅直径七英寸半的月球地图。这是第一幅使用精确测量的环形山位置的月球地图。事实上，梅耶测量了24个环形山的位置，并将其纳入地图中，使用测微计在纬度和经度上达到1'的精度。他还制作了一幅详细的瑞士地图，显示了十三个州，并准确标出了城镇、河流、湖泊和政治区划。梅耶的地图由Homann于1751年出版。梅耶通过引入许多制图学改进展示了他杰出的能力，但他还发现了月球天平动，并于Kosmographische Nachrichten Ⓣ（《宇宙志新闻》）（纽伦堡，1750年）中发表。这一成就和其他科学成就为他赢得了声誉，使他在1751年被任命为哥廷根经济学和数学教授。作为一名地图制作者，梅耶很自然地制作了一幅说明他从纽伦堡到哥廷根旅程的地图。这条经过班贝格和迈宁根的路线被认为是德国制作的第一幅道路地图。从他在哥廷根任职开始，梅耶就与莱昂哈德·欧拉通信。关于这些通信的详情，见[3]。

梅耶于1752年发明了对反射圆的一种巧妙改进。Joseph de Mendoza Rios在1801年提交给Philosophical Transactions of the Royal Society的论文《On an improved reflecting circle》中对此进行了详细描述。De Mendoza Rios写道：-

由于海上使用的反射仪器由手持支撑，其重量和刻度被限制在狭窄的范围内；似乎很难通过任何权宜之计来消除因尺寸小而产生的不便，而增大尺寸又是不可能的。然而，著名的梅耶设计了一种方法，在一次读数中确定所观测简单角度的倍数，而不是简单角度本身；通过这种方式，就上述误差而言，该仪器在实践中能够达到任何精度。他的发明与单纯重复观测有本质不同...

事实上，梅耶对反射圆的改进由让-夏尔·德博尔达进一步发展，并被让·巴蒂斯特·约瑟夫·德朗布尔和皮埃尔·梅尚用于子午线弧的测量，以努力定义米。

1754年，梅耶被任命为哥廷根天文台台长，他在那里继续工作直至去世。在哥廷根期间，他讲授数学、力学和光学，并将射影方法引入天文学和地理学。1755年3月1日，梅耶向哥廷根科学院发表演讲。他的演讲用拉丁语进行，题为《De transmutatione figurarum rectilinearum in triangula》Ⓣ（《将直线图形变换为三角形》）。在这次演讲中，他给出了一种确定不规则多边形面积的方法，该方法与他10年前书中提出的方法不同。Eric Forbes在[8]中提出了一个猜想：-

人们普遍认为他这次演讲的手稿已经遗失；但它可能从未存在过。考虑到由于与印刷商的不幸纠纷，《哥廷根评论》的出版已被暂停，而且梅耶能说一口流利的拉丁语，因此推测他可能决定以保存在哥廷根大学图书馆中他未发表著作里的一份德语论著为基础进行演讲，并非不合理。

梅耶于1753年开始计算月球和太阳表，并于1755年将它们寄给了英国政府。这些表足够精确，可以在海上以半度的精度确定经度。梅耶通过月距确定经度的方法以及一个用于校正由大气折射引起的经度误差的公式，在他去世后于1770年发表。

在1760年为他所编的表撰写的一篇序言中，梅耶说：-

我更加不愿意让我的表再被隐藏下去；尤其是因为几乎每个时代最著名的天文学家都热切希望有一种完美的月球理论……由于它在航海中的独特用途。我构建了这些表……关于运动的不均匀性，是依据伟大的艾萨克·牛顿的那个著名理论，那位杰出的数学家Eulerus首先将其优雅地归结为一般解析方程。

在《航海年鉴》的第一期中，有内维尔·马斯基林对梅耶的表的描述：-

已故的哥廷根教授梅耶已将月球表提高到足够的精确度，可以在海上将经度确定到一度以内，这一点从几位使用过这些表的人的试验中可以看出。必要计算的困难和长度似乎是阻碍它们被普遍使用的唯一障碍。

经度委员会送给 梅耶 的遗孀3000英镑作为这些表的奖励。尽管这是一大笔钱，但它远少于为解决经度问题而提供的奖金数额。詹姆斯·布拉德利，像 梅耶 一样，在编制月球表上投入了大量工作，他告诉 弗瑞兹·约翰 Harrison，若不是 Harrison 的“该死的表”，他和 梅耶 本可以分享那10000英镑的奖金。

梅耶编制了恒星表，并首次研究了80颗恒星的自行。他还编制了双星表。从1779年开始，即梅耶去世十七年后，威廉·赫歇尔开始寻找双星。他编制了269对双星表，其中227对是他首次发现的。然而，当他得到梅耶去世后出版的回忆录时，他发现梅耶发现了31颗他忽略了的双星。梅耶还有其他成就值得我们提及，例如他的地震理论、大气折射理论、磁学理论、视觉理论和颜色理论。一个花费许多时间仔细观测天体的人对视觉理论感兴趣，特别是对非常暗淡物体的视觉极限感兴趣，这并不奇怪。梅耶在1755年写道：-

……存在某个视角，低于此角度时，呈现给眼睛的物体要么不够清晰，要么甚至完全不清晰，只是模糊不清，仿佛从视线中消失。……我们将称此角度为视觉极限，并将通过实验研究其角度。……在此类情况下看到的物体，除非它们在眼睛中张角大于34''，否则将不可见，张角更小的物体肯定逃过视觉敏锐度。

最后，让我们简要看看梅耶的颜色理论。它包含在他给哥廷根科学院的一次讲座中，题为De affinitate colorum commentatio Ⓣ（关于颜色亲和性的评论）。他的目标，类似于他对视觉极限的研究，是试图确定眼睛能够区分多少种颜色。他以红、蓝、黄为三种基本颜色，然后研究混合这些基本颜色的若干十二分之一部分。他假设小于十二分之一的部分眼睛不可见。例如，他会声称19份红对1份蓝与红色无法区分，但11份红对1份蓝眼睛可以与红色区分。使用这个12份规则，他然后构造了诸如绿色（6份蓝，6份黄）的颜色。这导致他构造了一个“颜色三角形”，其顶点是他的三种基本颜色，以及91种不同颜色。然后他考虑添加最多四份黑或白，从而构造一个三维的颜色三角形阵列，每个三角形具有不同份数的黑或白。最终他得到了910种不同颜色，他声称眼睛可以区分。梅耶的颜色三角形在他去世时未发表，但兰伯特利用了该三角形并建议发表，这发生在1775年。事实上，兰伯特在1761年也利用了梅耶的恒星数据，当时他在其Cosmological Letters中给出的宇宙理论中使用了梅耶的具有自行的恒星表。

Tobias Mayer was a self taught mathematician. Although born in Marbach, he was brought up in Esslingen where he lived in very poor conditions. His first published work was Neue und allgemeine Art, alle Aufgaben aus der Geometrie vermittelst der geometrischen Linien leichter insbesondere wie alle reguläre und irreguläre Vielecke, davon ein Verhältnis ihrer Seiten gegeben, in den Circul geometrisch sollen eingeschrieben werden, sammt einer kurzen nötigen Buchstaben-Rechnenkunst und Geometrie Ⓣ which was published in Esslingen in 1741. Let us note that he was only eighteen years old when this book was published, a remarkable achievement for anyone but particularly for someone from a poor family. Eric Forbes writes in [8]:-

Tobias Mayer's interest in mathematics appears to have stemmed from his reading of the two books: Christian von Wolff's 'Anfangs-Gründe aller mathematuscher Wissenschaften' Ⓣ and Johann Christian Sturm's 'Mathesis enucleata' Ⓣ. At any rate, he acknowledges his debt to these two books when, on the occasion of his eighteenth birthday, he wrote the preface to his first printed work - a treatise concerned with the application of algebraic methods to problems of elementary and higher geometry. A comparison of Mayer's book with those others indicates that he was dependent upon Wolff for his knowledge of mathematical techniques and upon Sturm for providing the model on which he based his own presentation. The method which is given by the former for determining the area of an irregular polygon (or irregularly-shaped field) was illustrated by Mayer in his next major work, the 'Mathematischer Atlas' Ⓣ (Augsburg, 1745). ...

He was employed as a cartographer in Nürnberg from 1746, when he began working for the Homann Heirs company which succeeded the firm established in Nürnberg by the famous cartographer Johann Baptist Homann. In 1749, while working for the Homann Company, Mayer produced a map of the moon measuring seven and a half inches in diameter. It was the first map of the moon which used accurately measured positions of the craters. In fact Mayer measured the positions of 24 craters, which he included in the map, using a micrometer to obtain an accuracy of 1' in latitude and longitude. He also produced a detained map of Switzerland, showing the thirteen Cantons with accurately placed towns, rivers, lakes and political divisions. Mayer's map was published by Homann in 1751. Mayer showed his outstanding abilities by introducing many improvements in cartography, but he also discovered the libration of the Moon which he published in Kosmographische Nachrichten Ⓣ (Nürnberg, 1750). This and other scientific achievements gained him fame which led to his appointment as Professor of Economics and Mathematics at Göttingen in 1751. Being a map maker, it was natural for Mayer to make a map illustrating his journey from Nürnberg to Göttingen. The route, through Bamberg and Meiningen, is considered the first road map produced in Germany. From the time he took up his position in Göttingen, Mayer corresponded with Leonard Euler. For details of this correspondence see [3].

Mayer invented a clever improvement to the reflecting circle in 1752. It is described in detail by Joseph de Mendoza Rios who submitted his paper 'On an improved reflecting circle' to the Philosophical Transactions of the Royal Society in 1801. De Mendoza Rios writes:-

As the reflecting instruments employed at sea are supported by the hand, their weight and scale are limited within a narrow compass; and it seemed very difficult to obviate, by any expedient, the inconveniences arising from the smallness of their size, while it was impossible to increase it. The celebrate Tobias Mayer contrived, however, a method to determine, at one reading, instead of the simple angle observed, a multiple of the same angle; and, by this means, the instrument became, in practice, capable of any degree of accuracy, as far as regards the above mentioned errors. His invention is essentially different from the mere repetition of the observations ...

In fact Tobias Mayer's improvements to the reflecting circle were further developed by Jean Charle de Borda and used in the measurements of the arc of the meridian by Jean Baptiste Delambre and Pierre Méchain in their efforts to define the metre.

In 1754 Mayer was made Director of the Göttingen Observatory where he continued to work until his death. During his time in Göttingen, he lectured on mathematics, mechanics and optics, and introduced projective methods into astronomy and geography. On 1 March 1755, Mayer addressed the Göttingen Academy of Sciences. His address, given in Latin, was entitled 'De transmutatione figurarum rectilinearum in triangula' Ⓣ. In this address he gave a method for determining the area of an irregular polygon which was different from that presented in his book 10 years earlier. Eric Forbes makes a conjecture in [8]:-

It is generally believed that his manuscript of this lecture is lost; but it may never have existed. Bearing in mind that the publication of the Göttingen Commentarii had been suspended owing to an unfortunate dispute with the printer, and that Mayer spoke Latin eloquently, it is not unreasonable to conjecture that he may have decided to base his talk upon [a] German tract preserved among his unpublished writings in the Göttingen University Library.

Mayer began calculating lunar and solar tables in 1753 and in 1755 he sent them to the British government. These tables were good enough to determine longitude at sea with an accuracy of half a degree. Mayer's method of determining longitude by lunar distances and a formula for correcting errors in longitude due to atmospheric refraction were published in 1770 after his death.

In a preface written to his tables written in 1760 Mayer says:-

I am the more unwilling my tables should lie any longer concealed; especially as the most celebrated astronomers of almost every age have ardently wished for a perfect theory of the Moon ... on account of its singular use in navigation. I have constructed theses tables ... with respect to the inequalities of motions, from that famous theory of the great Newton, which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.

In the first issue of the Nautical Almanac there was a description by Nevil Maskelyne of Mayer's tables:-

The Tables of the Moon had been brought by the late Professor Mayer of Göttingen to a sufficient exactness to determine the Longitude at Sea to within a Degree, as appeared by the Trials of several Persons who made use of them. The Difficulty and Length of the necessary Calculations seemed the only Obstacles to hinder them from becoming of general Use.

The Board of Longitude sent Mayer's widow £3000 as an award for the tables. Despite this being a large amount of money, nevertheless it was much less that the amount on offer for solving the longitude problem. James Bradley, who had like Mayer, put a great deal of work into the production of lunar tables told John Harrison that he and Mayer would have shared the £10,000 prize money but for Harrison's "blasted watch."

Mayer catalogued stars and made the first study of the proper motions of 80 stars. He also made a catalogue of double stars. Beginning in 1779, seventeen years after Mayer's death, William Herschel began searching for double stars. He produced a catalogue of 269 such pairs, 227 of which were first discovered by him. However, when he was given the memoir of Tobias Mayer, published after his death, he found that Mayer had discovered 31 double stars that he had overlooked. There are other achievements by Mayer which we should mention such as his theories of earthquakes, atmospheric refraction, magnetism, vision, and colour. It is not surprising that someone who spent many hours in careful observations of astronomical objects should be interested in the theory of vision, and in particular the limits of vision for very faint objects. Mayer wrote in 1755 that:-

... there is a certain visual angle below which an object presented to the eye appears either not distinct enough or not even distinct at all, but only confused and as though it had vanished from sight. ... We shall call this angle the limit of vision, and we shall investigate its angle by experiment. ... objects seen under such circumstances will not be visible unless they subtend in the eye an angle of more than 34'', those subtending a smaller angle will definitely escape visual acuity.

Finally, let us look briefly at Mayer's theory of colour. It is contained in a lecture he gave to the Göttingen Academy of Sciences entitled De affinitate colorum commentatio Ⓣ. His aim, similar to his investigation of the limits of vision, was to attempt to establish how many colours the eye is capable of distinguishing. He took red, blue and yellow as the three basic colours, and then looked at mixing a certain number of twelfth parts of these basic colours. He postulated that amounts smaller than a twelfth part would not be visible to the eye. For example, he would claim that 19 parts red to one part blue was indistinguishable from red, but 11 parts of red to 1 part of blue was distinguishable by the eye from red. Using this 12 part rule, he then constructed colours such as green (6 parts blue, 6 parts yellow). This led him to construct a 'colour triangle' with his three basic colours at the vertices, and 91 different colours. He then considered adding up to four parts of black, or white, thereby constructing a three dimensional array of colour triangles each with a different number of parts of black or white. In the end he had 910 different colours which he claimed could be distinguished by the eye. Mayer's colour triangle was unpublished at his death but Johann Heinrich Lambert made use of the triangle and suggested that it be published, which took place in 1775. In fact Lambert also made use of Mayer's stellar data in 1761 when he used Mayer's catalogue of stars with proper motions in his theory of the universe given in his Cosmological Letters.