Fincke was born in Flensburg, Schleswig and died in Copenhagen. From 1577, he studied mathematics, rhetoric and other philosophical studies for five years at the University of Strasbourg. In 1590, he became professor of mathematics at the University of Copenhagen. In 1603 he also obtained a professorship in medicine. His lasting achievement is found in his book Geometria rotundi (1583), in which he introduced the modern names of the trigonometric functionstangent and secant.
Geometriae rotundi libri XIIII, 1583
His son in law was the Danish physician and natural historian, Ole Worm, who married Fincke's daughter Dorothea.