尼古拉斯·墨卡托Nicholas Mercator

尼古拉斯·墨卡托实际上叫Niklaus Kauffman。他后来改了姓，这在当时很常见，改为墨卡托，这是“商人”的拉丁语形式。墨卡托的父亲是Martin Kauffman，他从1623年起在荷尔斯泰因的奥尔登堡担任校长，直到1638年去世。尽管我们没有证据证明墨卡托上过他父亲的学校，但很难相信他会去上除他父亲任教的奥尔登堡学校以外的任何学校。

他于1632年进入罗斯托克大学，1641年获得学位，随后前往莱顿短期学习。1642年返回罗斯托克后，他被任命为该校哲学系的职位。1648年，墨卡托迁至哥本哈根大学。在那里工作期间，他出版了多部关于球面三角学、地理学和天文学的教科书[1]：-

……即使并非特别原创，它们也显示出他对要点的牢固掌握。

1651年出版了三部著作，即Trigonometria sphaericorum logarithmica Ⓣ（球面三角学与对数）、Cosmographia Ⓣ（宇宙志）和Astronomica sphaerica Ⓣ（球面天文学）。第一部列出了正弦、余弦、正切和余切函数在1°间隔上的对数，并展示了如何利用对数函数解三角形。第二部涉及自然地理学，而第三部则是关于球面天文学的著作。后续出版物还有：Rationes mathematicae subductae Ⓣ（数学理由的解释）（1653年）旨在区分有理数与无理数数，指出在音乐中有理比导致和谐，而无理比导致不和谐。他再次给出一个天文学例子，其中有理比对应于约翰内斯·开普勒以正多面体表述的行星系统结构，而无理比对应于观测到的运动。大概在同一年，墨卡托出版了De emendatione annua diatribae duae Ⓣ（年中日数分配的修正），其中他主张一种新的历法版本，其中有12个月，分别有29、29、30、30、31、31、32、31、31、31、30、30天。

在哥本哈根大学工作六年后，墨卡托因瘟疫导致大学关闭而不得不离开。从那时起，他的境况就不太好了。他十年间没有发表任何东西，但我们确实知道他一直保持着对天文学的兴趣。论文[2]考察了1659年和1660年初伊斯梅尔·布里阿德与墨卡托之间的三封信：-

伊斯梅尔·布里阿德，墨卡托，杰雷米亚·霍罗克斯：后开普勒天文学家的这一罕见组合出现在1659年和1660年初的简短通信中。伊斯梅尔·布里阿德与墨卡托之间交换的三封信，涉及杰雷米亚·霍罗克斯著名的凌日观测，象征性地连接了后开普勒思想的三个不同阶段。每位天文学家都对开普勒行星理论的接受、修改和传播做出了重要而独特的贡献。……伊斯梅尔·布里阿德虽然拒绝了约翰内斯·开普勒的面积定律和他对天体动力学的设想，但在十七世纪中叶以椭圆轨道的首要支持者而闻名；而他的年轻通信者墨卡托，多部数学和天文学著作的作者，则在本世纪后半叶主张开普勒椭圆和面积定律优于其他椭圆假说。

墨卡托在1660年前的某个时候去了英国，几乎肯定是为了改善自己的境遇。我们确实知道克伦威尔知道墨卡托关于修改历法的提议，一些历史学家认为克伦威尔可能邀请过墨卡托去英国。然而，这一假设没有直接证据。当然，墨卡托本希望获得大学职位，但未能找到这样的职位，于是在伦敦做私人辅导。然而，他很快在英国被公认为一位优秀的数学家，并开始与威廉·欧垂、约翰·佩尔和约翰·柯林斯交流思想。他十多年来的第一部出版物，也是他在英国的第一部，是Hypothesis astronomica nova（1664年在伦敦出版），其中他将约翰内斯·开普勒的椭圆轨道理论与自己的其他想法结合起来。很可能正是通过阅读这部著作，艾萨克·牛顿了解到约翰内斯·开普勒关于行星轨道是椭圆的主张。

当时的一大难题是能够确定海上的经度。一个已知的理论解决方案是设计一台在海上能保持准确时间的钟。墨卡托发明了这样一种航海天文钟，一种摆钟，凭借这项发明，他于1666年11月被选为皇家学会的会士。他还为皇家学会测量了气压，并结识了罗伯特·胡克，两人有许多共同的兴趣。

墨卡托发现了著名的级数，有时称为Mercator's series，

$\ln(1 + x) = x - \large\frac{1}{2}\normalsize x^{2} + \large\frac{1}{3}\normalsize x^{3} - \large\frac{1}{4}\normalsize x^{4} + ...$

他在1668年Logarithmotechnia中发表了这一结果。这个级数也被皮耶特罗·曼戈里研究过。

在接下来的一年，即1669年，应约翰·柯林斯和Seth Ward的请求，墨卡托将Kinckhuysen的荷兰文著作Algebra ofte Stel-konst Ⓣ（集合与数的代数）翻译成拉丁文。这使墨卡托与艾萨克·牛顿建立了联系，两人通信，除其他事项外还讨论了月球的运动。墨卡托于1676年出版了一部两卷本的天文学著作Institutiones astronomicae Ⓣ（天文学基础）。这些是写得很好的文本，涵盖了该学科的最新知识。艾萨克·牛顿读了这些著作，事实上艾萨克·牛顿的那本带有他阅读时所作笔记的副本仍然存在。

人们可能会认为墨卡托的声誉足以让他获得一个好职位，但看来他未能如愿。1676年，罗伯特·胡克提名墨卡托担任基督医院的数学教师职位，但他未被任命。由于无法获得教职而深感悲伤，他于1682年移居法国，这次是为了填补一个特定职位，即设计凡尔赛宫的水利工程。让-巴蒂斯特·柯尔贝尔，1666年在巴黎创立Académie des Sciences并担任法国财政总监，曾邀请墨卡托承担该项目。不幸的是，墨卡托与柯尔贝尔发生了争执，因此这个项目没有成功。

有理由将墨卡托与墨卡托混淆，因为墨卡托也研究过墨卡托的地图投影。

Nicolaus Mercator was actually called Niklaus Kauffman. He later changed his family name, which was a common thing to do at this time, to Mercator which was the Latin form of 'merchant'. Nicolaus's father was Martin Kauffman who was a school master at Oldenburg in Holstein from 1623 until his death in 1638. Although we have no evidence to prove that Nicolaus attended his father's school, it is impossible to believe that he would attend any other than the one in Oldenburg where his father taught.

He entered the University of Rostock in 1632, received a degree in 1641, then went to Leiden for a short period. After his return to Rostock in 1642 he was appointed to a post in the Faculty of Philosophy at the University. In 1648 Mercator moved to the University of Copenhagen. While he was working there, he published a number of textbooks on spherical trigonometry, geography and astronomy [1]:-

... if not markedly original, they show his firm grasp of essentials.

Three texts were published in 1651, namely Trigonometria sphaericorum logarithmica Ⓣ, Cosmographia Ⓣ, and Astronomica sphaerica Ⓣ. The first tabulates logarithms of the sine, cosine, tangent and cotangent functions at 1° intervals and shows how to solve triangles using logarithmic functions. The second of these texts deals with physical geography, while the third is a text on spherical astronomy. Further publications followed: Rationes mathematicae subductae Ⓣ (1653) sets out to distinguish between rational and irrational numbers pointing out that in music rational ratios lead to harmony while irrational leads to dissonance. Again he gives an astronomical example where rational ratios corresponds to Kepler's structure of the planetary system given in terms of regular polyhedra, while irrational ratios correspond to the observed motions. Probably in the same year Mercator published a text De emendatione annua diatribae duae Ⓣ in which he argues for a new version of the calendar in which there are 12 months with 29, 29, 30, 30, 31, 31, 32, 31, 31, 31, 30, 30 days respectively.

After working at the University of Copenhagen for six years, Mercator had to leave when the university was closed due to the plague. From this time on things were not too good for him. He published nothing for ten years but we do know that he maintained his interests in astronomy. The paper [2] looks at three letters between Boulliau and Mercator in 1659 and early 1660:-

Ismael Boulliau, Nicolaus Mercator, Jeremiah Horrocks : this rare conjunction of post-Keplerian astronomers occurs in a brief exchange of letters in 1659 and early 1660. Three letters exchanged between Boulliau and Mercator, concerning the famous transit observation of Horrocks, symbolically link three distinct phases of post-Keplerian thought. Each astronomer made significant, but distinctive, contributions to the acceptance, modification, and transmission of Keplerian planetary theory. ... Boulliau, although rejecting Kepler's area rule and his vision of a celestial dynamics, became known in the middle decades of the seventeenth century as the leading proponent of elliptical orbits; whereas his younger correspondent, Mercator, author of several mathematical and astronomical texts, argued the superiority of the Keplerian ellipse and area rule over rival elliptical hypotheses in the latter half of the century.

Mercator went to England some time before 1660, almost certainly in an attempt to improve his fortunes. We do know that Cromwell was aware of Mercator proposals for changing the calendar and some historians suggest that Cromwell may have invited Mercator to England. There is no direct evidence for this hypothesis, however. Certainly Mercator would have liked to have received a university appointment but, unable to find such a position, he did private tutoring in London. He was soon recognised in England, however, as a fine mathematician and he began to exchange ideas with Oughtred, Pell and Collins. His first publication in over ten years, and his first in England, was Hypothesis astronomica nova (published in London in 1664) in which he combined Kepler's theory of elliptical orbits with other ideas of his own. It was probably through reading this text that Newton learned about Kepler's claim that the orbits of the planets were ellipses.

One of the big problems of the day was being able to determine the longitude at sea. A known theoretical solution was to devise a clock which would keep accurate time at sea. Mercator invented such a marine chronometer, a pendulum clock, and on the strength of this invention he was elected a Fellow of the Royal Society in November 1666. He also made measurements of air pressure for the Royal Society and got to know Hooke with whom he shared many common interests.

Mercator discovered the well known series, sometimes called Mercator's series,

$\ln(1 + x) = x - \large\frac{1}{2}\normalsize x^{2} + \large\frac{1}{3}\normalsize x^{3} - \large\frac{1}{4}\normalsize x^{4} + ...$

He published this in Logarithmotechnia 1668. This series was also investigated by Mengoli.

In the following year of 1669, at the request of Collins and Seth Ward, Mercator translated the Dutch text Algebra ofte Stel-konst Ⓣ by Kinckhuysen into Latin. This brought Mercator into contact with Newton and the two men exchanged letters, discussing among other things the motion of the moon. Mercator published a further two volume work in astronomy Institutiones astronomicae Ⓣ in 1676. These were well written texts covering the latest knowledge on the subject. Newton read these works and in fact Newton's copy containing the notes he made as he read it still exists.

One might have thought that Mercator's reputation was sufficient that he would have been able to acquire a good post but it seems that he could not. In 1676 Hooke proposed Mercator for the post of Mathematical Master at Christ's Hospital, but he was not appointed. Saddened by his inability to get a teaching position, in 1682 he moved to France, this time to fill a specific position, namely to design the waterworks at Versailles. Jean-Baptiste Colbert, founder of the Académie des Sciences in Paris in 1666 and controller general of finance in France, had invited Mercator to undertake the project. Unfortunately Mercator fell out with Colbert so this project was not a success.

There is some reason to confuse Nicolaus Mercator with Gerardus Mercator since Nicolaus also worked on Gerardus's map projection.