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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 middleclass 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 (15321584), a wealthy welleducated 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 part of 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 (15321600), 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 twentysix 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 (15361614) and Theodor Zwinger (15331588) at Basel, and Hieronymus Fabricius (15371619) and Giulo Casserio (15451605) 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 welltravelled versatile scientist with great expertise in mathematics, astrology, astronomy and medicine, he was very wellrespected and came to the attention of the leading men such as Duke Philipp von HolsteinGottorf who appointed him as the personal physician to his court in 1590. On 30 August 1590, Fincke married Söst Ivers (15741614) at the Ducal Palace in Tonder. Fincke was twentynine 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 Sorensen, 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 (159899, 160607, 161516, 162425 and 163334). He continued to lecture at the University until he was eightyone 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 triangutorum.
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 (15921663), went on to become Professor of Mathematics and Physics at the University of Copenhagen. Anna (15941677), 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 (15951665), Dorothea (15961628) and Drude (16041671) all married university professors. When Thomas Fincke died he left seventynine descendants: children, grandchildren, greatgrandchildren and greatgreatgrandchildren. These descendants dominated the academic professors at the University of Copenhagen during the 17^{th} 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.
Article by: J J O'Connor and E F Robertson
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 Mathematicians born in the same country

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