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Mathematical Treasures - On Triangles, by Regiomontanus

Frank J. Swetz and Victor J. Katz

This is the title page of On Triangles by Regiomontanus (Johannes Müller) (1436-1476).  Although the work was written in 1464, it was not published until 1533.  The page can be translated as follows:

Five Books on Triangles of Every Kind, by the most learned man and extraordinary professor of the mathematical disciplines, Johannes Regiomontanus:  In which is explained all things necessary for one who wishes to reach perfection in his knowledge of the astronomical sciences.  Since these matters have never been developed elsewhere before this time, one will aspire in vain to learn these things without these ideas.

An appendix contains the works of Nicolas of Cusa on the quadrature of the circle and the measuring of straight and curved lines, together with their never before published refutation by Johannes Regiomontanus.

All of these have been thoroughly edited with singular fidelity and diligence at the publishing house of John Petreus, in Nürnberg, A.D. 1533.

A complete translation and analysis of On Triangles is available in Barnabas Hughes, editor and translator, Regiomontanus on Triangles (Madison: University of Wisconsin Press, 1967).


This page contains theorems 12 (XII) and 13 (XIII) of Book II of On Triangles (De triangulis):

Theorem 12:  If the perpendicular is given and the base and the ratio of the sides are known, then each side is known.

Theorem 13:  If each of the segments of the base is known and also the ratio of the sides, then each side and the perpendicular may be found.

There is a calculation in the margin by an owner of this book.  It would be interesting to work out the details.

Index to Mathematical Treasures

Frank J. Swetz and Victor J. Katz, "Mathematical Treasures - On Triangles, by Regiomontanus," Convergence (January 2011)

Mathematical Treasures from the Smith and Plimpton Collections at Columbia University