汤玛斯·芬克Thomas Fincke

汤玛斯·芬克的名字出现时有多种不同拼写，最常见的是我们采用的这个，但除了拉丁语形式的Finchius和Finckius之外，有时也使用Finke、Finck或Fink。他的父母，Raadmand Jacob Fincke和Anna thor Smede（或Thorsmede），来自弗伦斯堡最受尊敬的家庭。Anna来自一个富裕的丹麦中产阶级家庭，但遗憾的是，她在儿子芬克出生六天后去世。芬克的父亲被描述为一位“博学的商人”，他曾是Martin Luther和Philipp Melanchthon的学生，后者是维滕贝格的希腊语教授，也是Luther最早的支持者之一。本传记的主人公芬克以他的祖父命名，祖父也叫芬克，曾任弗伦斯堡市长。当本传记的主人公芬克只有九岁时，他的父亲去世，从那时起他就与叔叔埃拉斯穆斯·赖因霍尔德 thor Smede（1532-1584）一起生活，后者是一位富有的、受过良好教育的当地议员。

芬克于1570年至1577年间在弗伦斯堡一所新创办的路德宗拉丁语学校就读，当时他16岁。他在这所学校接受了良好的广泛教育，学习了路德的宗教思想，主要使用⟦N4⟧编写的教科书，并学习了希腊语、拉丁语和希伯来语等古代语言。然而，他对数学所知甚少，离开学校时，除了基本的算术技能外，他对这门学科一无所知。弗伦斯堡位于石勒苏益格，虽然由丹麦管理，但那里有大量讲德语的人口，并与附近的德国城市有着密切的联系。腓特烈二世于1559年至1588年担任丹麦国王，在七年战争后，他于1570年与瑞典签署了和平条约，确保瑞典保持独立。腓特烈二世在战后重建国家，于1571年授予石勒苏益格居民与丹麦所有其他居民同等的权利，因此当芬克离开学校时，他本可以前往哥本哈根大学在这所一流学府学习。如果他选择就读德国城市的一所大学，人们会期望他去相对较近的罗斯托克。然而，可能是受到曾在斯特拉斯堡和维滕贝格学习的叔叔的影响，他决定前往更远的南方，并于1577年在斯特拉斯堡学院注册入学。鉴于他在弗伦斯堡学校接受的人文主义教育，人们不会期望他在学院学习数学，因为他在这门学科上几乎没有任何基础。然而，⟦N4⟧坚信，那些打算在大学学习神学、法律或医学的人，应该首先熟悉数学所特有的精确思维方式。因此，数学是芬克学习的主要科目之一。

他在斯特拉斯堡花了五年时间学习数学、占星术、哲学和修辞学。他在斯特拉斯堡科学院的数学教授是Conrad Dasypodius（1532-1600），因其在1571年至1574年间对斯特拉斯堡大教堂天文钟的研究而闻名。Dasypodius是一位优秀的数学家，曾出版过欧几里得的Elements以及诸如Analyseis geometricae sex librorum Euclidis Ⓣ（对欧几里得几何六卷的分析）（1566年）和Mathematicorum disciplinarum principia Ⓣ（数学原理）（1573年）等文本。Finke在斯特拉斯堡科学院师从Dasypodius学习期间几乎肯定研究过这些著作。离开科学院后，芬克于1582年2月入读海德堡大学，在那里读了一个学期后，又于1582年夏季在莱比锡大学读了一个学期。他还在耶拿大学和维滕贝格大学度过了一段时间，之后于1583年春天遵照其监护人赖因霍尔德 thor Smede的指示返回弗伦斯堡。此次返回的原因是在收到父亲遗嘱留下的一大笔遗产后处理其财务事务。然而，瘟疫的爆发使他在弗伦斯堡逗留的时间远超预期，直到六个月后的1583年秋天，他才离开家乡前往巴塞尔。正是在那里，他出版了Geometriae rotundi libri XIIII （1583年），这是他在斯特拉斯堡着手的一项工作，确保了他作为数学家的声誉。我们将在下文讨论这项工作，但在此之前，我们将继续描述他的职业生涯。

在发表了如此重要的数学著作之后，人们可能会认为芬克会继续取得更大的数学成就。然而，他的目标仍然是从事医学职业，他最初的数学学习主要是为了遵循⟦N4⟧的教导。他现在开始在巴塞尔学习医学，并于1583年11月在帕多瓦继续学习。他在帕多瓦度过了四年，期间短暂访问了其他大学，即1585年的锡耶纳和1586年的比萨。芬克在帕多瓦留下了深刻的印象，尽管他离开时只有二十六岁，但他在建立医学图书馆方面的成就受到了高度赞赏，以至于当地教堂“圣索菲亚”中竖立了一根刻有纪念牌的大理石柱以纪念他。他回到巴塞尔，提交了医学博士学位论文，并于1587年8月24日成功获得学位。他的医学老师是巴塞尔的⟦N6⟧（1536-1614）和⟦N7⟧（1533-1588），以及帕多瓦的⟦N8⟧（1537-1619）和⟦N9⟧（1545-1605）。然而，他并没有完全放弃对数学和天文学的兴趣，因为他在帕多瓦期间与乔凡尼·安东尼奥·马吉尼友好，并且在获得博士学位后回到家乡弗伦斯堡后，他继续与乔凡尼·安东尼奥·马吉尼通信。

芬克没有直接从巴塞尔返回弗伦斯堡，而是继续在奥地利、匈牙利和波希米亚进行考察旅行，然后才在家乡弗伦斯堡开设了医疗诊所。作为一名游历广泛、多才多艺的科学家，他在数学、占星术、天文学和医学方面拥有丰富的专业知识，备受尊敬，并引起了像荷尔斯泰因-戈托普公爵菲利普这样的显要人物的注意，后者于1590年任命他为宫廷私人医生。1590年8月30日，芬克在滕德尔的公爵宫与Söst Ivers（1574-1614）结婚。芬克结婚时二十九岁，而Söst当时刚过完十六岁生日。Jürgen Schönbeck在[6]中写道：-

毫无疑问，此时芬克无论在职业上还是私人生活上都处于他成功人生的巅峰。

大约在他结婚的时候，芬克收到了校长Niels Kaas的邀请，请他担任哥本哈根大学数学讲席，Niels Kaas是芬克的赞助人之一。1590年11月菲利普公爵的去世使他的决定变得稍微容易一些，次年7月，他接替Anders Krag担任该职位。1602年2月，他成为哥本哈根大学的修辞学教授，填补了1月27日Hans Guldsmed去世后空缺的讲席。1602年7月28日，哥本哈根大学医学教授Peder Sørensen去世，次年4月，芬克被任命填补这一讲席。除了这些学术职位外，他还在哥本哈根大学担任过许多重要职务。他于1593年和1599年担任哲学系主任，从1603年起担任医学院院长长达53年，直到1656年去世，并五次担任大学校长（1598-99、1606-07、1615-16、1624-25和1633-34）。他一直在大学讲课直到八十一岁，但即使在那之后，他仍然保留着官方职位和收入。这意味着他是一个非常富有的人，能够提供奖学金并向贵族家庭贷款。

他最著名的书Geometriae rotundiⓉ（几何旋转）（1583年）是作为教科书编写的，读者可参考约翰·缪勒了解更多细节。该书基于彼得吕斯·拉米斯的著作，他从那里借用了标题中的“rotundi”一词，意为圆和球，以及“radius”一词。这本书引入了“切线”和“割线”这两个术语，芬克设计了新的公式，如正切定理。这部用拉丁文写成的著作分为14卷，但将它们视为章节更为合适。圆的基本理论在第1至4卷中介绍，平面三角学在第5至11卷中研究，最后三卷讨论球面三角学。14卷中的大多数都有副标题：

第一卷。De circuli rectis secantibusⓉ（将圆向右切割）。

第二卷。De lineis circuli tangentibus sigillatim & coniunctim cum secantibus, atque obliquis Ⓣ（关于与圆相切的切线所画出的、以及斜的，密封而紧密的联合）。

第三卷。

第四卷。De cruribus anguli in semicirculo Ⓣ（关于半圆中其腿的角，）。

第五卷。De semicirculi sinibus, tangentibus, secantibus Ⓣ（关于半圆中的正弦、正切、正割）。

第六卷。De adscriptione circuli & trianguli Ⓣ（关于圆与三角的成员资格）。

第七卷。De adscriptione triangulati Ⓣ（三角化的成员资格）。

第八卷。De tetragonismo circuli Ⓣ（化圆为方）。

第九卷。De canone triangulorum Ⓣ（三角学准则）。

第十卷。De calculo triangulorum Ⓣ（论三角形的计算）。

第十一卷。De Geodaesia rectarum Ⓣ（论测地线）。

第十二卷。

第十三卷。De triangulis sphaericis Ⓣ（论球面三角形）。

第十四卷。De calculo triangulorum sphaericorum Ⓣ（论球面三角形的计算）。

这部著作仅有400多页，但其中近三分之一是三角函数表，印在第9卷和第10卷之间。用于标准三角函数的缩写芬克标志着今天通用记法的开端。对于“正弦”，他写作“sin.”；对于“余弦”，他写作“sin. com.”；对于“正切”，他写作“tan.”；对于“正割”，他写作“sec.”；对于“余割”，他写作“sec.com.”；对于“余切”，他写作“tan.com.”。芬克的书得到了克里斯托佛·克拉乌、约翰·纳皮尔和巴塞洛缪·皮提斯卡斯的推荐，他们都从中采纳了许多内容。他关于天文学和占星术的其他著作则远没有那么引人关注，尽管他曾与第谷·布拉赫和约翰内斯·开普勒有联系。这些著作包括： Horoscopographia, continens fabricam cardinum coelestium ad quodvis datum tempus: et viam deductionis Ptolemaicam Ⓣ（Horoscopographia，即容器建筑在任何给定时间摇动天体，以及托勒密体系的推演）（1585年），以及Horoscopographia sive de inveniendo stellarum astrologia situ Ⓣ（或寻找恒星，天文学状况Horoscopographia）（1591年）。

最后，让我们简要考察一下芬克和Söst Finckes的大家庭，以及他们对丹麦学术家族许多代人的影响。他们的长子Jacob Fincke（1592-1663）后来成为哥本哈根大学的数学和物理学教授。Finckes家女儿中的长女Anna（1594-1677）嫁给了Caspar Bartholin，一位医生和神学家，而著名数学家拉斯穆·巴多林是他们六个儿子中的第二个。Finckes家的女儿Margaret（1595-1665）、Dorothea（1596-1628）和Drude（1604-1671）都嫁给了大学教授。芬克去世时留下了七十九名后代：子女、孙辈、曾孙辈和玄孙辈。这些后代在17世纪主导了哥本哈根大学的学术教授职位。

让我们以引用Jürgen Schönbeck [6]给出的总结来结束这篇传记：-

芬克……是十七世纪丹麦最重要、最杰出的科学家之一，一位处于现代科学初期的数学家、占星家和医生，一位人文主义的代表和一位有影响力的学术组织者。

Thomas Fincke's name appears with a variety of different spellings, the commonest is the one we have adopted but Finke, Finck, or Fink are sometimes used in addition to the Latin versions of Finchius and Finckius. His parents, Raadmand Jacob Fincke and Anna thor Smede (or Thorsmede), came from the most respected families in Flensburg. Anna was from a rich middle-class Danish family but, sadly, she died six days after her son Thomas was born. Thomas's father was described as a "learned merchant" who had been a student of Martin Luther and Philipp Melanchthon, the Professor of Greek in Wittenberg and one of Luther's first supporters. Thomas, the subject of this biography, had been named after his grandfather, also named Thomas Fincke, who had been mayor of Flensburg. When Thomas, the subject of this biography, was only nine years old, his father died and from that time he lived with his uncle Reinhold thor Smede (1532-1584), a wealthy well-educated local councillor.

Thomas attended a newly founded Lutheran Latin school in Flensburg from 1570 to 1577 when he was 16 years old. He received a good broad education at this school, being taught Luther's religious ideas, mainly from textbooks written by Philipp Melanchthon, and taught the ancient languages of Greek, Latin and Hebrew. He learnt little of mathematics, however, and when he left the school he would have had no knowledge of the subject beyond basic arithmetical skills. Flensburg was in Schleswig and, although administered from Denmark, it had a large German speaking population and had strong links with nearby German cities. Friedrich II was king of Denmark from 1559 to 1588 and, after the Seven Years' War, had signed a peace treaty with Sweden in 1570 ensuring that Sweden remained independent. Friedrich II, building up his country after the war, in 1571 granted inhabitants of Schleswig equal rights with all other inhabitants of Denmark so that when Fincke left school, he had could have gone to the University of Copenhagen to study at this top class establishment. If he had chosen to attend a university in one of the German cities one would have expected him to study at the relatively nearby Rostock. However, probably influenced by his uncle who had studied at Strasbourg and Wittenberg, he decided to go very much further south and, in 1577, he matriculated at Strasbourg Academy. With the humanist education that he had at school in Flensburg one would not expect him to study mathematics at the Academy since he had hardly any background in the subject. However, Melanchthon firmly believed that those intending to study theology, law or medicine at university, should first become familiar with the precise way of thinking that was characteristic of mathematics. As a consequence, mathematics was one of the major subjects that Fincke studied.

He spent five years studying mathematics and astrology, philosophy and rhetoric at Strasbourg. His mathematics professor at Strasbourg Academy was Conrad Dasypodius (1532-1600), famed for his work on Strasbourg Cathedral's Astronomical Clock between 1571 and 1574. Dasypodius was a fine mathematician and had produced editions of Euclid's Elements and texts such as Analyseis geometricae sex librorum Euclidis Ⓣ (1566), and Mathematicorum disciplinarum principia Ⓣ (1573). Finke would almost certainly have studied these works while studying under Dasypodius at the Strasbourg Academy. After leaving the Academy, Fincke enrolled at the University of Heidelberg in February 1582 and then, after a semester there, he enrolled for a semester at the University of Leipzig in the summer of 1582. He spent some time at the universities of Jena and Wittenberg before returning to Flensburg in the spring of 1583 on the instructions of his guardian Reinhold thor Smede. The reason for this visit was to sort out his financial affairs after receiving a substantial legacy from his father's will. However, an outbreak of the plague meant that he remained much longer at Flensburg than he had intended and only after six months, in the autumn of 1583, did he leave his hometown and go to Basel. It was there that he published Geometriae rotundi libri XIIII (1583), a work he had undertaken at Strasbourg, which has guaranteed his fame as a mathematician. We discuss this work below but before doing so we will continue to describe his career.

Having published such an important mathematical work, one might assume that Fincke would go on to still greater mathematical achievements. However, his aim was still to follow a career in medicine and his preliminary study of mathematics had largely been undertaken to follow the teaching of Melanchthon. He now began his study of medicine in Basel, continuing it Padua in November 1583. He spent four years in Padua, interspersed with short visits to other universities, namely Siena in 1585 and Pisa in 1586. Fincke made a big impression in Padua and, although he was only twenty-six years old when he left, his achievements in founding a medical library were so highly appreciated that a marble pillar with inscribed commemorative plaque was put in the local church 'Santa Sofia' in his honour. He returned to Basel where he submitted his thesis for a medical doctorate and was successful in obtaining the degree on 24 August 1587. His medical teachers had been Felix Plater (1536-1614) and Theodor Zwinger (1533-1588) at Basel, and Hieronymus Fabricius (1537-1619) and Giulo Casserio (1545-1605) at Padua. He did not entirely leave behind his interest in mathematics and astronomy, however, for he was friendly with Giovanni Antonio Magini while at Padua and, after he returned to his home town of Flensburg after the award of his doctorate, he continued to correspond with Magini.

Certainly Fincke did not return directly to Flensburg from Basel but rather made further study trips through Austria, Hungary and Bohemia before settling into a medical practise in his home town of Flensburg. Being a well-travelled versatile scientist with great expertise in mathematics, astrology, astronomy and medicine, he was very well-respected and came to the attention of the leading men such as Duke Philipp von Holstein-Gottorf who appointed him as the personal physician to his court in 1590. On 30 August 1590, Fincke married Söst Ivers (1574-1614) at the Ducal Palace in Tonder. Fincke was twenty-nine years old when he married Söst the day after her sixteenth birthday. Jürgen Schönbeck writes in [6]:-

Undoubtedly at this time Fincke was, both professionally and privately, at the high point of his successful life.

Around the time of his marriage, Fincke received an offer of the Chair of Mathematics at the University of Copenhagen from Chancellor Niels Kaas, one of Fincke's patrons. The death of Duke Philipp in November 1590 made the decision somewhat easier for him and, in July of the following year, he took up the post succeeding Anders Krag. He became professor of rhetoric at Copenhagen in February 1602, filling the chair which became vacant on the death of Hans Guldsmed on 27 January. On the 28 July 1602, Peder Sørensen, the professor of medicine at Copenhagen died and in April of the following year Fincke was appointed to fill this chair. In addition to these academic posts, he held a number of important roles at the University of Copenhagen. He was Dean of the Faculty of Philosophy in 1593 and 1599, he was Dean of the Medical School for 53 years from 1603 until his death in 1656 and five times he was Rector of the University (1598-99, 1606-07, 1615-16, 1624-25 and 1633-34). He continued to lecture at the University until he was eighty-one years of age, but even after this he still retained his official position together with his income. This meant he was a very wealthy man and was able to offer scholarships and lend money to noble families.

His most famous book Geometriae rotundi Ⓣ (1583), was intended as a textbook and the reader is referred to Regiomontanus for more details. It was based on works by Peter Ramus from whom he took the word 'rotundi' of the title, meaning circle and sphere, and the word 'radius'. The book introduces the terms 'tangents' and 'secants' and Fincke devised new formulae such as the law of tangents. The work, written in Latin, is divided into 14 Books but it would be more appropriate to think of them as chapters. The elementary theory of the circle is presented in Books 1 to 4, plane trigonometry is studied in Books 5 to 11, and the final three Books treat spherical trigonometry. Most of the 14 Books have subtitles:

Liber Primus. De circuli rectis secantibus Ⓣ.

Liber Secundus. De lineis circuli tangentibus sigillatim & coniunctim cum secantibus, atque obliquis Ⓣ.

Liber Tertius.

Liber Quartus. De cruribus anguli in semicirculo Ⓣ.

Liber Quintus. De semicirculi sinibus, tangentibus, secantibus Ⓣ.

Liber Sextus. De adscriptione circuli & trianguli Ⓣ.

Liber Septimus. De adscriptione triangulati Ⓣ.

Liber Octavus. De tetragonismo circuli Ⓣ.

Liber Nonus. De canone triangulorum Ⓣ.

Liber Decimus. De calculo triangulorum Ⓣ.

Liber Undecimus. De Geodaesia rectarum Ⓣ.

Liber Duodecimus.

Liber Decimustertius. De triangulis sphaericis Ⓣ.

Liber Decimusquartus. De calculo triangulorum sphaericorum Ⓣ.

The work consists of just over 400 pages, but almost a third of these pages are taken up with trigonometric tables which are printed between Books 9 and 10. The abbreviations Fincke used for the standard trigonometric functions mark the beginning of the notation which is current today. For 'sine' he wrote 'sin.', for 'cosine' he wrote ' sin. com.', for 'tangent' he wrote 'tan.', for secant he wrote 'sec.', for 'cosecant' he wrote 'sec.com.', and for 'cotangent' he wrote 'tan.com.'. Fincke's book was recommended by Clavius, Napier and Pitiscus all of whom adopted much from it. His other works on astronomy and astrology are of much less interest despite the fact that he was in touch with Brahe and Kepler. These include: Horoscopographia, continens fabricam cardinum coelestium ad quodvis datum tempus: et viam deductionis Ptolemaicam Ⓣ (1585), and Horoscopographia sive de inveniendo stellarum astrologia situ Ⓣ (1591).

Finally let us look briefly at the Thomas and Söst Finckes' large family and the influence they had on the academic families of Denmark for many generations. Their eldest son, Jacob Fincke (1592-1663), went on to become Professor of Mathematics and Physics at the University of Copenhagen. Anna (1594-1677), the eldest of the Finckes daughters, married Caspar Bartholin, a physician and theologian, and the famous mathematician Erasmus Bartholin was the second of their six sons. The Finckes' daughters Margaret (1595-1665), Dorothea (1596-1628) and Drude (1604-1671) all married university professors. When Thomas Fincke died he left seventy-nine descendants: children, grandchildren, great-grandchildren and great-great-grandchildren. These descendants dominated the academic professors at the University of Copenhagen during the 17th century.

Let us end this biography but quoting the summary given by Jürgen Schönbeck [6]:-

Thomas Fincke ... was one of the very most important and significant scientists in Denmark during the seventeenth century, a mathematician and astrologer and physician in the beginning of modern science, a representative of humanism and an influential academic organizer.