Mathematical Treasures - Peter Apianus's Trigonometry and Geography

Frank J. Swetz and Victor J. Katz


Petrus Apianus (Peter Apian) was a German humanist known for his work in mathematics, astronomy, and cartography. This is the title page of his Instrumentum primi mobilis (1534), which discusses the use of measuring instruments and presents new developments in the compiling of trigonometric tables. This book is also noteworthy in that it contains the first European printing of the Latin translation by Gerard of Cremona of the revised edition of Ptolemy's Almagest by Jabir ibn Aflah (fl. 12th century). Jabir is commonly known in the West by the Latinized version of his name, "Geber".



This page of Instrumentum directly precedes a listing of sine values. Apian illustrated and explained the nomogram employed to graphically determine his sine and versed sine values (versed sine of arc ø = 1 - cos ø).



Title page of Petrus Apianus’s A Geographical Introduction (1534). In this book, Apian reviewed the theories of Vernerus, or Johannes Werner (1468-1522), a Nuremburg priest and mathematician who devised a method of using lunar observations to find longitude, and explained applications of trigonometry (specifically, sines and chords) in geography.


Convergence's Mathematical Treasures also include images from Apian's 1533 Folium populi: Instrumentum and from his 1545 Cosmographia, the latter extended from his original 1524 edition by his student Gemma Frisius.


Index to Mathematical Treasures